UC-NRLF $B 3Db 371 Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/firstbookinarithOOcalirich CALIFORNIA STATE SERIES SCHOOL TEXT-BOOKS FIRST BOOK IN iVRITHlVEETIC COMPir.ED BV THE SXAXE XEXX-BOOK COMMITTEE AND APPROVED BY THE STATE BOARD OF EDUCATION SACRAMENTO: >UPE IS- W. W. SHANNON, SUPERINTENDENT STATE PRINTING Copyright, 1905, by THE STATE OF CALIFORNIA Copyright, 1905, by J. W. McCLYMONDS and D. R. JONES. Entered at Stationers' Hall, London. EDUOAXrON -DEPT , In the compilation of this work certain matter from McCly- tnondt anil Jones's Blementary Arithmetic has been VM«d. All •wcA matter is protected by the copyright entry nottd abovt. 6e— 50M— 2, '10 PREFACE It is quite generally admitted that the results obtained in the field of primary arithmetic are by no means com- mensurate with the attention which the subject receives in our schools. After several years of earnest effort, the pupils too often show little or no insight into number relations. Furthermore, the ability to handle numbers with accuracy and a fair degree of facility is also wanting. This lack of results, we believe, is due chiefly to the inade- quacy of the methods commonly pursued in the teaching of this subject. It too frequently occurs that beginners are introduced to meaningless abstractions (which are retained only with difficulty, notwithstanding their fre- quent repetition), and are hurried on to the fundamental operations, which are rarely well mastered. • The work outlined in Chapter I of this book is designed to prepare the pupils for the intelligent mastery of the fundamental operations as presented in the succeeding chapters. Through the application of number to objects, an insight into number relations and the common opera- tions is gained. Throughout this chapter the memorizing of facts is subordinate to the getting of ideas. The mastery of the fundamental operations is taken up in the succeeding chapters, and a well-developed method is provided for each operation. The general plan of work is a simple one. The pupils are required to memorize a M187502 4 PREFACE few number facts and to apply these until they have be- come perfect reflexes, before new facts are introduced. A constant review is provided, as the facts of previous lessons are involved in the drill exercises which follow the several lessons. Simple work in fractions and denominate numbers is introduced in the first lessons, and continued throughout the book. A balance is maintained between the mechani- cal work on the one hand and the solution of problems on the other. The problems are of a practical character, and are drawn largely from the field of everyday experience. Whenever a problem of a given type seems to offer a lan- guage difficulty, several such problems are given in suc- cession. Such repetition frequently occurs in the earlier pages of the text. Frequent requests for detailed information with refer- ence to the methods herein contained, and favorable reports from teachers and superintendents who have found these methods helpful in their work, have led to the preparation of this text, which, we trust, will be found of service in its field of intended usefulness. CONTENTS CHAPTER I Preliminary Type Lessons PAGE Simple Directions — Magnitude — Counting — Comparison of Quantities — Grouping — Writing Numbers — Division by Measurement and Partition — Measurements — Forms — Fractions — Difference — Number Stories — Measure of Time — Summary 7-33 CHAPTER II General Introduction Steps in Addition, Subtraction, Multiplication, and Division 34-42 CHAPTER III Addition and Subtraction Notation — Numeration — Addition — Subtraction — Objective Fractions — Compound Numbers — Multiplication — Divi- sion 43-114 CHAPTER IV Multiplication and Division Addition — Subtraction — Multiplication — Division — Simple Fractions — Compound Numbers 115-187 CHAPTER V Common and Decimal Fractions Written Fractions — Decimals — Compound Numbers — Per- centage — Interest 188-244 6 CONTENTS CHAPTER VI Denominate Numbers ' PAGE Tables — Time Problems — Measure of Length — Square and Cubic Measure — Lumber Measure — Cash Account — Angles — Roman Notation 245-256 ELEMENTARY ARITHMETIC CHAPTER I PRELIMINARY LESSONS To THE Teacher. The lessons of this chapter are designed to indicate the nature of the work that should be done by the class before the text is placed in the hands of the pupils. Space does not permit of the introduction of sufficient material to furnish all of the exercises which the pupils will need. The lessons in this chapter should, there- fore, be regarded as type lessons which, the teacher is to expand and to supplement to meet the needs of the class. It is not expected that all of the work suggested in any one of the lessons will be given at any one time. Several parallel lines of work are suggested in the various lessons, and these should be carried on together, the work gradu- ally increasing in difficulty until the pupils have finally mastered all of the work indicated in each of the lessons. No abstract number work, aside from counting and the reading and writing of numbers, is provided, and none should be given. The pupils should deal with number in its relation to things, and not with abstract number facts. In the exercises suggested in this chapter, the pupils them- selves play an important part. They are required to c?o, as well as attend to what is done by others. They are led to discover number relations in the quantities that are 7 g " * *'* PREJJMINARY LESSONS handled by them, and to express these reiations in correct language. These lessons are presented in the form of questions and directions given by the teachers, as serving best to illustrate the methods to be followed in presenting them to the class. No exercise should be continued so long that the pupils will begin to lose interest in it. The teacher is expected to use whatever objects may be at hand, and to vary the objects frequently. A box of 1-inch cubes will be found very useful in this as well as in subse- quent number work. They are easily handled, and may be conveniently arranged to show relative quantities, etc. The pupils should be encouraged to express themselves freely, but at the same time correctly. Sufficient time should be taken to acquaint the pupils with the language forms involved in these lessons. If properly presented, the work suggested in this chapter should not only give tlie pupils familiarity with simple number facts, and an insight into the common operations with number, but should also establish habits that will be found extremely valuable in all subsequent number work. Some of the number facts developed in these lessons are designated as facts to be learned. These should be perfectly memorized. LESSON I — MAGNITUDE The primary purpose of Lesson I is to train the pupils to hear and interpret simple directions. The secondary purpose is to acquaint the pupil with the language forms used to denote relative position, direction, magnitude, etc. After giving a direction, allow sufficient time for all the pupils to interpret the direction, and to image its execution. Should the pupil called upon fail to execute correctly the MAGNITUDE 9 direction given, do not call upon a second pupil for it, but give a new direction. This will be found more effec- tive in keeping the attention of the whole class than the more common procedure, namely, that of permitting the brighter pupils to do most of the work. Later, return to the direction upon which a failure was made, and call upon some other child to carry it out, unless you have reason to think that the pupil who once failed can now execute it correctly. Simple directions are to be executed, showing the mean- ing of the following and similar terms : in your right hand^ in your left hand, to the right of^ to the left of above, below, nearer^ farther from, beside, between, in front of, larger than, taller tha7i, tallest, shorter than, shortest, smallest, twice as lo7ig as, one half as long as, ttvice as far from, one half as far from, the same distance from, a line^ at the end of in the middle of etc. Illustration. Have two boys from the class, say John and James, stand in line in front of the class, John several feet to the right of James. Problems: 1. I want some one to stand in line with John and James. 2. Stand in line with the two boys, to the right of John. 3, Stand in line^ to the left of James. 4. Stand in line, so John is to your right and James is to your left. 5. Stand in line, the same distance from the boys. 6. Stand in line, nearer John than James. 7. Stand in line, farther from James than from John. 8. Stand in line, one half as far from John as from James. 9. Stand in line, to the right of John, one half as far from John as from James. 10. Stand out of line, but the same distance from each of the two boys. Illustrati(3N. With colored crayon draw on the board lines of different lengths, for comparison. Problems: 1. Alice, tell me a story about the yellow line and the 10 PRELIMINARY LESSONS blue line. Story : The yellow line is longer than the blue line. 2. Walter, tell me a story about the red line and the blue line. Story : The red line is shorter than the blue line. 3. Mary, tell me a story about the green line and the red line. Story : The green line is as long as the red line. 4. Who can tell me a story about the orange line and the blue line ? Story : The orange line is one half as long as the blue line. 5. Who can tell me a story about the yellow line and the red line ? Story : The yellow line is two times (or twice') as long as the red line. 6. Ethel, tell me a story about the red line, the blue line, and the yellow line. Story: The red line and the blue line together are as long as the yellow line. Substitute lines drawn with white crayon and lettered a, ^, 82 ADDITION AND SUBTRACTION 76. Study Exercises. Study as indicated in Steps A and B, pp. 48, 49. 7 6 7 9 4 9 6 2 9 8 To THE Teacher. Give oral drill on the com- binations in the above study. See p. 49. 77. Oral Exercises. Add each column as indicated in Step C, p. 49. a 6 c a e / 9 A I i fc I m n 4 7 4 9 7 6 7 6 9 9 9 9 6 4 9 4 9 7 6 7 4 7 7 1 1 7 9 9 7 9 7 4 7 7 9 7 6 9 9 6 1 7 6 7 4 9 7 6 7 4 7 1 1 9 7 6 7 6 9 7 4 9 8 9 2 9 9 1 1 5 9 6 9 2 8 7 4 9 7 9 7 6 8 1 78. Find the amount of the following bill : Oakland, Cal., June 5, 1905. Mr. T. H. Crane, Bought of Horace Mann & Co. 2 doz. eggs . . @$.20 40 2 lb. ham . . . @ .20 40 4 lb. butter . . @ .25 1 00 12 lb. sugar . . @ .05 60 4 lb. steak . . . @ .15 60 1 cabbage . 05 SUBTRACTION — LESSON E 83 SUBTRACTION — LESSON E 79. 1. Memorize the followmg : 16 9 18 12 12 16 9 12 -9 -2 -9 -6 -8 -7 — 7 -4 77964928 2. Give a number story suggested by each. 3. What must be added to 9 to make 16 ? 4. A class worked 16 examples m addition and subtraction. Seven of the examples were in subtrac- tion. How many of them were in addition ? 5. There are — months in a year. 6. If a boy attends school 8 months each year and has a vacation the remaining months, he has — months' vacation each year. 7. A boy bought a dozen bananas. He ate 4 .of them. He had — bananas left. 8. Nine inches and — inches are 18 inches. 9. A farmer had 16 sheep. He sold 7 of them. He had — sheep left. 10. A grocer sold 4 cans of corn from a- box con- taining a dozen cans. There were — cans left in the box. 11. A girl bought a dozen cookies. She gave away all but 4. How many did she give away ? 12. On Arbor Day the pupils planted 9 trees. Two of the trees died. How many of them lived? 84 ADDITION AND SUBTRACTION 80. Study Exercises. 16 12 9 12 16 9 18 12 -7 -6 -2 -8 -9 -7 -9 -4 Study the above until you can give the results readily. 81. Written Exercises. a h c d e 1. 9626 9828 9222 9826 5622 -1859 -1859 -6738 -6877 -634 2. 9308 9633 1843 8354 8923 -2529 -1638 -975 -2866 - 5237 3. 9386 8926 9540 6345 2873 -6499 -4180 -6814 -2378 -916 82. Add : Three hundred seventy dollars and twenty cents, forty -seven dollars and forty-five cents, one hundred four dollars and eighty cents, sixty-eight dollars and forty-seven cents, one dollar and sixty-eight cents, three hundred fifty-two dollars and thirty-seven cents, sixty dollars and eight cents. 83. Oral Exercises. a b c d e / 9 h i J 77 65 28 43 26 84 96 47 79 57 92 64 34 59 24 64 97 83 24 40 MULTIPLICATION — LESSON A 85 MULTIPLICATION — LESSON A 84. 1. Find the sura of a column of three 2's. 2. Find the sum of a column of three 3's. 3. Two 3's are — . Two 2's are — . 4. Three 2's are — . Three I's are — . 5. Two O's are — . Three O's are — . 6. How many 2's are there in 4 ? In 6 ? 7. How many 3's are there in 9 ? In 6 ? 3 8. Two 3's are 6 may be written thus : x 2 9. Read and memorize : ^ 3 4 2 2 3 2 x2 x^ x^ x3 x^ xjt ~6 "^ 4 6 9 8 10. The sum of 43 and 43 may be found by mul- tiplication, thus : Model : 43 Two 3's are 6 ; two 4's are 8. X 2 The answer is 86. 86 85. Multiply: ^ a b c d e / 9 h 1. 23 42 30 41 14 34 24 40 x2 x2 x2 x2 x2 x2 x2 x2 2. 234 403 312 231 203 123 212 120 x2 x2 x3 x3 x3 x3 x4 x4 The answer in multiplication is called the product. 86 ADDITION AND SUBTRACTION 86. 1. One half of 6 is — . One half of 10 is — . 2. Four is J of — . Five is ^ of — . 3. Four pints are — quarts, or gallon. 4. Four quarts are — pints. Six pints are — quarts. 5. Two yards are — feet. Nine feet are — yards. 6. Two nickels are — dimes. Two dimes are — cents. 7. One dollar is — cents. One dollar is — dimes. One half-dollar is — cents, or — dimes. 8. Six dimes are — cents. There are — half-dol- lars in one dollar. 9. There are — quarter-dollars in one dollar. 10. There are — quarter-dollars in one half-dollar. 11. There are — nickels in one quarter-dollar. 12. How many nickels make 20 cents? 30 cents? 13. How many half-dollars make 2 dollars ? 87. Oral Exercises. Add each column as indicated in Step C, p. 49. a b C d e / 9 h I J k I m n 7 9 8 8 3 2 3 3 4 G 5 4 2 8 6 4 2 7 9 5 7 4 6 6 5 4 3 7 5. 2 6 9 7 9 9 7 9 6 5 4 3 9 9 4 7 7 4 6 2 9 6 C 5 4 3 9 7 8 5 6 9 6 7 6 7 6 5 4 3 7 4 7 7 2 7 8 5 8 2 6 5 4 3 9 8 9 8 4 6 4 3 5 8 6 5 4 3 4 WRITTEN PROBLEMS 87 88. Written Problems. 1. A man bought a bicycle for $45 and a gun for $38. He sold both for $100. Find the amount of gain or loss. 2. A boy earns $10 a month and spends $6. How much will he save in 2 months? 3. What must be added to $75 to make $120 ? 4. A boy had 65 cents. How much money had he left after paying 25 cents for a ticket to a circus and 10 cents for some popcorn ? ^ 5. A girl read 54 pages of a book on Saturday and 25 pages on Sunday. The book contained 102 pages. How many more pages has she to read to finish the book ? 6. Mary picked 16 quarts of berries on Monday, 25 quarts on Tuesday, 17 quarts on Wednesday, and 8 quarts on Thursday. How many quarts did she pick in all ? 7. At 30 cents each, how much will 2 readers cost ? 89. Written Exercises. a 6 c d e / 9 h 544 737 774 269 643 309 398 232 289 446 336 441 946 721 716 796 364 655 776 269 153 809 406 749 613 737 334 441 975 941 688 959 707 983 776 434 985 969 926 474 829 795 362 239 733 259 686 656 88 ADDITION AND SUBTRACTION 90. 1. One whole is — thirds. 2. One whole is — sixths. One half is — sixths. 3. One third is — sixths. Two thirds are — sixths. 4. One half and one third are — sixths. 5. One half of a pie and one third of a pie are — sixths of a pie. 6. Two thirds and one half are — sixths. 7. One half of one third is — sixths. 8. Which is the larger, one half or one third ? One third or one sixth ? One half or three sixths ? One half or two sixths ? Two thirds or three sixths ? 9. One half and one fourth are — fourths. 10. The ratio of 2 to 4 is ; of :| to J is ; of |- to ^ is . 11. The ratio of 4 to 2 is — ; of ^ to ^ is — ; of •| to l is — . 91. Memorize: 1. One half and one fourth are three fourths. 2. One half and one third are five sixths. ADDITION — LESSON F 89 ADDITION — LESSON F 92. 1. Memorize the following : 7 3 5 9 9 _7 ^ J_ _1 _l 14 7 12 11 10 2. Give a number story suggested by each. 3. What is the sum of $ 7 and $ 7 ? 4. How many days are there in 2 weeks ? 5. Two 7's are — . One half of 14 is — . 6. A boy spent $ 4 for a suit of clothes and $ 3 for a pair of shoes. He spent $ — in all. 7. A boy had 10 words to spell. He missed one word. He spelled — words correctly. 8. Twelve months are 7 months and — months. 9. How much more is the sum of 7 and 7 tlian the sum of 7 and 5 ? 10. The sum of 9 and 2 is one more than the sum of 9 and — . 11. If one sheep costs $ 7, what will be the cost of 2 sheep ? 12. If a boy had 12 oranges and gave away 5 of them, he would have — oranges left. 13. Four days and — days make one week. 14. A ruler 12 inches long is cut into 2 pieces. If one piece is 7 inches long, the other piece is — inches long. 90 ADDITION AND SUBTRACTION 93. Study Exercises. Study as indicated in Steps A and B, pp. 48, 49. 7 3 5 9 9 7 4 7 2 1 7 2 7 1 4 5 9 7 9 3 To THE ; Teacher. Give oral drill. See p. 49. 94. Oral Exercises. Add as in Step C, p. 49. a b c d e / 9 h i J k 5 7 9 3 9 7 9 9 5 9 4 9 7 9 4 9 7 5 3 9 9 9 9 5 9 7 9 3 9 7 5 9 5 9 3 5 9 4 7 3 3 5 3 5 7 9 9 9 5 7 4 4 7 3 4 5 9 7 2 9 3 1 7 4 3 7 2 5 9 1 4 9 3 DIVISION — LESSON A 95. 1. Two 2's are — . Two 3's are — . Three 2's are — . Three 3's are — . Four 2's are — . Two 4's are — . 4 2. The number of 2's in 8 may be shown thus : 2)8 3. Bead and memorize : 1 1 1 A 1 1 2)6 2)4 3)6 2)8 3)9 4)8 The answer in division is called the quotient. SUBTRACTION — LESSON F 91 SUBTRACTION — LESSON F 96. 1. Memorize the following : 12 14 7 11 10 12 7 11 10 ^7 _7 -3 _2 -9 -5 -4 -9 -1 574917329 2. Give a number story suggested by each. 3. A girl picked 7 boxes of cherries. She sold all but 3 boxes. How many boxes did she sell ? 4. Some boys bought a dozen lemons. They used 7 of them in making lemonade. How many lemons had they left ? 5. A line 7 inches long is — inches shorter than a foot rule. 6. There are — days in 2 weeks. Two 7's are — . 7. How many less than 12 apples are 7 apples ? 8. How many less than 11 weeks are 9 weeks ? 9. Seven days are how many more than 3 days ? 10. Fourteen inches are how many more than 7 inches ? 11. How much longer than the sum of 4 inches and 3 inches is one foot ? 12. How much longer than the sum. of 2 inches and 9 inches is one foot ? 13. How many hours are there from 5 o'clock to 12 o'clock? 92 ADDITION AND SUBTRACTION 97. Study Exercises. 12 7 11 10 14 12 11 7 10 -7 _3 -9 -1 -7 -5 ^2 -4 -9 Study the above until you can give the results without hesitation. 98. Written Exercises. 1. a 7474 -3737 b 7272 -2727 C 7242 -3465 d 2010 -1999 e 7111 -3182 a 8401 -2749 b 9123 -3145 c 9445 -5677 d 7452 -4567 e 3572 -816 99. Study Exercises. 2)6 2)4 3)6 2)8 3)9 4)8 Study the above until you can give the answers readily. The number of 2's there are in 64 may. be found 32 thus : ^T-^ There are three 2's in 6, and two 2's in 4. There are thirty-two 2's in 64. 100. Divide-. a b c d e 1. 2)46 2)62 2)80 2)64 3)63 2. 2)6420 2)2604 2)4026 2)4602 2)2064 3. 3)3690 3)6309 3)'9603 3p»96 3)"63()9 ORAL PROBLEMS 98 101. Oral Problems. 1. How many pints are there in 3 quarts ? 2. How many yards are there in 9 feet ? 3. Three dollars are how many half-dollars ? 4. Eight pints are how many quarts ? 5. Two yards are — feet. Two gallons are — quarts. 6. A half-dollar is — dimes ; — nickels ; — cents. 7. Eighteen cents are — dime and — cents. 8. Six quarts are — gallon and — quarts. 9. The number of 2 cents there are in 6 cents may be shown thus : _§_ 10. Read and give quotients : 2^)47 3^)6? 4^)8? 2^)87 $3)p $2)p 11. Read and divide : 3ft.)66ft. $4)$48 $2)$80 2 in.)60 in. 102. Oral Exercises. Add each column as indicated in Step C, p. 49. a 6 C d e / g /« I J k z m W 4 9 7 6 3 2 8 8 7 4 5 8 6 7 6 2 9 8 8 6 6 7 4 9 6 4 4 8 7 6 9 7 7 9 2 8 7 6 8 5 6 7 6 4 8 8 6 7 8 6 4 7 6 9 8 9 8 6 7 7 8 4 7 7 6 1 7 2 6 7 5 7 2 8 7 4 3 7 3 4 4 6 9 9 7 9 5 5 7 8 4 2 5 5 9 6 9 3 94 ADDITION AND SUBTRACTION 103. 1. We can find one half of 8 books by separating the books into — equal groups. 2. Show one half of 6 books ; of 8 books. 3. One half of 8 books is — books. This may be shown thus : 4 books 2)8 books 4. Read and give quotients : 2)4 books 3)6 books 2)$ 8 2)$ 6 3)9 ft. 5. Read and divide : abed 2)$ 286 2)$ 402 2)$ 840 2)608 ft. 3)360 da. 6. Find 1 of $460. Find 1 of $390. - 7. Find i of $408. Find l of 680 pounds. 8. A man had $84. He spent one half of it for a wagon. How much did the wagon cost him ? 104. Written Exercises. a b C d e / 9 h 547 923 897 246 875 141 646 574 656 759 265 463 336 676 246 306 878 386 682 675 474 519 346 759 697 779 376 978 385 434 346 686 874 525 449 926 673 608 346 853 382 467 772 437 469 187 346 418 492 487 839 299 843 515 346 707 WRITTEN PROBLEMS 95 105. Written Problems. 1. After selling 47 sheep a farmer had left 38 sheep. How many sheep had he at first ? 2. A farmer had 32 cows. He bought 29 more cows. How many cows had he then ? 3. Of a school of 436 pupils, 169 are boys. How many girls are there in the school ? 4. Find the cost of 3 cows at $ 23 each. 5. There are 55 pupils in the First Grade, 46 pupils in the Second, and 37 pupils in the Third. How many pupils are there in the three grades ? 6. There are 35 girls and 18 boys in a school. How many more girls than boys are there in the school ? 7. A boy had 48 marbles. He sold one fourth of them. How many did he sell ? How many marbles did he have left ? 8. A grocer bought flour at 89 cents a sack and sold it at $ 1.00 a sack. How much did he make on each sack ? 9. A horse that cost $ 86 was sold at a gain of $ 18. Find the selling price. 106. Solve : a 6 c d e / $234 $403 $320 432 ft. 201 ft. 233 yd. x2 x2 x3 x2 x4 x3 ft. ft. yd, 96 ADDITIOISr AND SUBTRACTION 9 • • , • • • ••• (a) (b) (c) 107. Divide group a into two equal parts; group h into three equal parts ; group c into six equal parts. Let each dot represent a pupil. 1. — pupils are ^ of 6 pupils. 2. — pupils are J of 6 pupils. 3. — pupil is ^ of 6 pupils. 4. Six pupils are — times 3 pupils. 5. Six pupils are — times 2 pupils. 6. How many 2 pupils are there in 6 pupils ? 7. How many 3 pupils are there in 6 pupils ? 8. Six pupils are how many times 2 pupils ? 9. Six pupils are how many times 3 pupils ? 10. One half of 6 pupils is — more pupil than one third of 6 pupils. 11. The ratio of 3 pupils to 6 pupils is — — ; of 2 pupils to 6 pupils is . 12. The ratio of 6 pupils to 3 pupils is — ; of 6 pupils to 2 pupils is — ; of 6 pupils to one pupil is — . 13. The difference between J of 6 pupils and -^ of 6 pupils is — pupil. 108. Add: H 5* 9^ 2i H H ii H 2i H ADDITION — LESSON G 97 ADDITION — LESSON G 109. 1. Memorize the following : 8 9 6 7 9 _8 _6 _5 1 _1 16 15 11 8 17 2. Give a number story suggested by each. 3. What is the sum of 9 and 8 ? 4. What is the sum of two 8's ? ■ 5. A boy has 9 marbles in one pocket, and 6 in the other. How many marbles has he in both pockets ? 6. Seventeen is — more than 8. 7. Two 8's are 16. One half of 16 is — . 8. What is the ratio of 16 to 8 ? Of 8 to 16 ? 9. Fred has 6 pigeons. Walter has 5 pigeons more than Fred. How many pigeons has Walter ? 10. A piece 5 ft. long was sawed from a board 11 ft. long. How long was the part that remained ? 11. A post 8 ft. long is 1 ft. below ground. How long is the part above ground ? 12. A boy delivers 6 quarts of milk each morning and 5 quarts each evening. How many quarts does he deliver each day? 13. From a board 16 ft. long a piece 8 ft. long is cut. How long is the part remaining ? 1st ]\k Aimth _7 98 ADDITION AND SUBTRACTION 110. Study Exercises. Study as indicated in Steps A and B, pp. 48, 49. 8 9 6 7 9 8 6 5 18 8 18 5 6 9 7 8 6 9 To THE Teacher. Dictate for oral addition the combinations studied in the above exercises. See p» 49. 111. Oral Exercises. Add each column as indicated in Step C, p. 49. a 6 C d e / g h i j k I m n 9 1 6 6 5 6 1 8 9 8 9 9 9 5 7 9 9 9 8 9 9 1 7 7 8 1 1 8 6 7 8 8 7 8 7 9 5 5 1 9 9 7 9 6 1 7 6 1 5 7 1 8 9 1 1 5 8 9 9 5 6 8 8 6 7 9 6 8 8 6 8 6 9 9 8 5 9 1 8 8 9 8 113. Write in a column and add: 1, 138.67, $.88, $67.46, $.89, $69.34. 2. $85.89, $.70, $8.05, $67.96, $9.77. 113. Write and solve : 1. $24.93 -$8.15. 4. $30.00 -$6.44, 2. $104.50 -$7.15. 5. $90 -$17.50. 3. $70.42-$5.79. 6. $85.46-118. 15 11 17 8 -9 -6 -9 -1 6 5 8 7 s SUBTRACTION — LESSON G 99 SUBTRACTION — LESSON G 114. 1. Memorize the following : 15 11 17 8 16 -6 -5 -8 -7 -8 9 6 9 18 2. Give a number story suggested by each. 3. A man earns $15 a week and spends $6 a week. He saves $ — each week. 4. A post 11 feet long stands in a hole 5 feet deep. How much of the post is above ground ? 5. A milkman sold 8 quarts of milk from a can containing 17 quarts. How many quarts remained in the can ? 6. A grocer sold 15 lb. of sugar in two packages. If one of the packages weighed 9 lb., how much did the other weigh ? 7. Mary is 8 years old and her sister is 17 years old. Mary is — years younger than her sister. 8. Ethel went to visit Lottie on the ninth of June and stayed until the fifteenth of June. How long was her visit ? 115. Multiply: ah c d e f 423 3132 2012 4023 1022 3012 2 3 4 2 4 3 100 ADDITION AND SUBTRACTION 116. Study Exercises. 15 11 8 17 16 15 8 11 17 _9_5_^ _8 zl Z^ zl IL^ zl Study the above until you can give the results without hesitation. 117. Written Exercises. a b c d e / 1. 8515 8717 1755 8655 1556 1777 -849 -6859 -886 -7759 -858 -878 2. 1715 5365 9537 9608 7474 1423 -849 - 1578 -2769 -6809 -1837 -348 3. 9847 6328 8456 9167 3135 8636 -2038 - 1885 -3778 -5569 -439 -2747 MULTIPLICATION — LESSON B 118. Show by addition and multiplication : 1. The sum of two 5's ; of four 3's ; of three 4's. 2. The sum of five 2's : of two 6's ; of six 2's. 119. Bead and memorize : 1. 5 4 2 3 x2 x3 x5 x5 To 12 lo T5 6 3 2 • 5 x2 x4 x6 x3 12 12 12 15 2. What is the answer in multiplication called? MULTIPLlCATIOfN-'-.LKSO:^^' h 101 120. Study Exercises. 5 4 2 5 6 3 2 x2 x3 x5 x3 x2 x4 x6 x5 Study the above until you can give the results without hesitation. Give the products in the above exercises from right to left, adding 1, 2, and 3 to each product, thus : 15, 16 ; 12, 13 ; 12, 13 ; etc. 121. Written Exercises. Model for Exercise a: 465 Carry in multi- 2 plication as in ad- 930 dition. o 5 c d e / (1 7i 1. 405 365 260 345 250 123 203 332 x2 x2 x2 x3 x3 x4 x4 x-4 2. 210 122 201 231 102 405 604 213 X 5 X 5 x6 x5 x6 x3 x2 403 x4 3. 345 234 222 320 1.32 305 413 x2 x3 x6 x4 x5 x2 x2 x3 122. Oral Exercises. Add: b C d e / h i 70 69 75 53 58 43 36 07 25 70 ^- 61 86 45 64 59 94 83 76 102 ADDITION ANO SUBTRACTION DIVISION — LESSON B 123. 1. Count by 3's to 18 ; by 2's to 24 ; by 4's to 20 ; by 5's to 30. 2. Arrange 6 books to show that 2 books are ^ of 6 books. 3. One third of 6 books is — books. Two thirds of 6 books are — books. 4. Draw oblongs to represent 9 books. One third of 9 books is — books. Two thirds of 9 books are — books. 5. Show J of 12 squares. Show | of 12 squares. 6. Three fourths of 12 squares are — squares. 7. Show 1 of 12 circles. Show ^ of 12 circles. 124. Bead and memorize : 43352562 3)12 5)15 4)12 3)15 6)12 2)10 2)12 5)10 125. Oral Exercises. Add as indicated in Step C, p. 49. a h C d e / 9 h i J A; I m n 3 6 4 5 4 7 3 9 7 5 9 8 4 3 7 8 7 6 G G 8 7 8 7 6 9 8 5 9 7 9 8 2 8 2 8 2 6 7 1 9 f> 7 8 8 5 3 i 7 9 5 9 6 9 7 8 5. 6 3 G 8 9 4 7 6 9 9 1 4 7 7 5 G 8 9 7 6 4 8 7 5 9 6 G 9 8 7 9 6 2 8 9 4 5 C 8 6 4 DIVISION — LESSON B 103 126. Study Exercises. 3)12 5)T5 4jTI 3)15 6)T2 2)l0 2)T2 5)T0 Study the above until you can give the quotients without hesitation. 1. 2)12 This means : How many 2'5 are there in 12? Or, What is ^ of 12 2 The answer is — . 2. 2)p2 This means: What is J of $12? The answer is — . 3. $2)$ 12 This means: Hoiv many $2 are there in $12? The answer is — . 127. Read and find quotients : a h c d e 1. 2)2648 4)8048 4)1248 2)1046 $3)$ 1296 2. 6)1206 5)T050 5)T005 4)1^08 3)6012 ft. 128. Written Exercises. a h c d 6 / g h i 799 999 788 899 949 997 978 889 998 617 899 476 937 994 556 292 789 871 696 991 768 946 794 958 926 875 358 949 946 489 479 893 779 374 965 663 978 999 987 558 794 497 898 899 994 538 426 259 538 369 838 366 429 439 129. Solve: a 6 C d $84.93 $43.21 $87.53 $97.34 -$17.16 -$14.52 -$48.79 -$17.36 104 AUDITION AND SUBTRACTION Halves Thirds Fourths Sixths Eighths 130. Show the truth of each statement by folding or cutting paper. 1. ^ of a pie is more than ^ of a pie. 2. ^ of a pie is less than ^ of a pie. 3. ^ of a pie + 1- of a pie is the same as -J of a pie. 4. ^ of a pie + ^ of a pie is more than -|^ of a pie. 5. f of a pie + i^ of a pie is ^ of a pie less than a whole pie. 6. I of a pie + J of a pie is ^ of a pie more than a whole pie. 7. 1 of a pie + ^ of a pie is ^ of a pie less than a whole pie. 8. I of a pie is J- of a pie less than ^ of a pie. 9. I of a pie + J of a pie is |- of a pie more than ^ of a pie. 10. If we cut :|- of a pie from | of a pie, there will remain ^ of a pie. 131. Memorize : a b c d h t 1 f i -i +i + i i i 5 8 i=H +i ADDITIOX — LESSON H 105 ADDITION — LESSON H 132. 1. Memorize the following : 1 5 7 2 8 8 9 9 4 1 14 11 3 3 11 2. Give a number story suggested by each. 3. What two combinations in this lesson give 11 as a sum ? 4. What must be added to $5 to make $14 ? 5. Name another combination whose sum is 14. 6. The sum of 9 and 9 is 18 ; of 9 and 8 is 17 ; of 9 and 7 is 16. When a number is added to 9, the sum ends in a figure one less than that added to 9. 7. A girl has 4 books of poems and 7 story books. How many books has she in all ? 133. Oral Exercises. The sign = between two quantities shows that they are equal in amount. 4 + 5 = 9. This means that the sum of 4 and 5 is equal to 9. 4 + 5 = 6 + 3. This means that the sum of 4 and 5 is equal to the sum of 6 and 3. 134. Supply the number that should stand in place of X. 1. 7 + 4 = 3+^'. 3. 9 + 7 = 8 + ;r. 5. 7 + 8 = 6 + a;. 2. 9 + 5 = 8 + ^:. 4. 6 + 5 = 4 + a:. 6. 5 + 9 = 7+ic. 106 ADDITION AND SUBTRACTION 135. Study Exercises. Study as indicated in Steps A and B, pp. 48, 49. 15 7 2 8 8 9 4 13 3 19 4 8 8 2 5 7 1 To THE Teacher. Dictate the combinations in- volved in the above study. See p. 49. 136. Oral Exercises. Add as indicated in Step C, p. 49. a b c d e f g hi j k I m 8 7 578288 9 8822 2-5 85278712787 7888852598 5 29 5728287812881 1582739893415 893148 5358729 137. Drill columns. A drill column is one in which a g combination occurs several times. To make a drill column n 'for 8 and 7: Write the combination at the foot of the ^ column. The sum is 15. Place in the column a number that will increase the sum to either 18 or 17. This number ^ is either 3 or 2. Either can be used. If 2 is taken, the sum 8 is increased to 17. Then place 8 in the column. The sum, 2 is 25. Again add either 3 or 2, and continue as above. n Write a drill column for 6 and 7; for 9 and 6; for 8 and 0; for, 9 and 7. 8 SUBTRACTION — LESSON H 107 SUBTRACTION — LESSON H 138. 1. Memorize thefolloioing : 9 14 11 9 3 11 14 11 11 3 -1 -5 -4 -8 -2 -8 -9 -7 -3 -1 8 9 7 113 5 4 8 2 2. Give a number story suggested by each. 3. A board 11 ft. long will make two shelves, one 4 ft. long and the other — ft. long. 4. How many days are there from April 3 to April 11? 5. There were 11 marbles in a ring. Frank shot 3 of them out of the ring. There were — marbles left in the ring. 6. A farmer sold 14 sacks of grain. Five were wheat, and the rest were oats. He sold — sacks of oats. Draw a diagram to show the places mentioned in each of the following problems : 7. Harry's home is 1 mile north of the school- house, and Willie's home is 2 miles south of the school- house. How far apart do they live ? 8. Mary's home is 4 blocks east of the school- house, and Edna's home is 7 blocks west of the school house. How far apart do they live ? 9. Fred lives 9 miles west of the city, and James lives 14 miles west of the city. How far apart do they live ? 108 ADDITION AND SUBTRACTION 139. Study Exercises. 9 14 11 9 3 11 14 11 11 8 -1 -5 zl ZL^ zl -8 -9 -7 -3 -1 Study the above until you can give the results without hesitation. 140. Written Exercises. a 1. 3841 -1687 6 9341 -944 C 3114 -275 d 3411 -2463 e 9141 -7398 / 9319 -1041 2. 6056 -3968 9327 -8469 2526 -783 8436 -4759 5143 -2436 9357 -6468 3. 9418 -4476 9473 -6627 9365 -7467 9785 -6489 3368 -2579 7991 -4898 141. Oral Exercises. Add each columr 1 as indicated in Step C, P' i9. a b c d e / h i j A- / m « 6 8 9 7 5 9 9 4 3 9 5 4 2 9 9 9 5 9 6 5 7 5 5 4 3 8 7 4 1 5 1 4 5 8 6 5 •5 6 7 2 3 6 6 7 4 8 6 7 9 8 9 7 9 8 9 5 4 S 6 2 4 3 7 2 1 3 1 2 1 8 9 5 4 6 5 7 3 8 5 7 9 8 i 9 6 7 9 8 6 7 8 5 9 4 3 8 9 ORAL PROBLEMS 109 142. Oral Problems. 1. One dollar is — cents ; — nickels ; - — dimes ; — half-dollars. 2. A half-dollar is — cents; — nickels; — dimes; — quarter-dollars. 3. A quarter-dollar is — cents; — nickels; 2 dimes and — cents. ^ 4. If Edna buys a box of berries for 15 cents and gives the clerk a 2o-cent piece, how much change will she receive ? 5. Mabel buys 35 cents' worth of sugar and gives the clerk a half-dollar. The clerk counts the change as he gives it to Mabel. He begins with the cost of the sugar and says, 35 and 5 are 40, and 10 are 50, as he gives her a nickel and a dime. 6. Ethel bought 30 cents' worth of ribbon and handed the dealer a half-dollar. Count the change. 143. Have the pupils take turns at ^' keeping store." Supply them with paper coins (or better, with real coins), and have them make purchases and count the change. Make change for : 1. 40 cents out of $1.00. 5. 30 cents out of $1.00. 2. 60 cents out of $1.00. 6. 15 cents out of $5.00. 3. $1.25outof $5.00! 7. $3.50 out of $10.00. 4. $2.25 out of $5.00. 8. $4.25 out of $5.00. 110 ADDITION AND SUBTRACTION 144. Oral Problems. 1. A boy paid 50^ for a baseball and 30^ for a glove. How much did he pay for both ? 2. Harry and James picked two boxes of apples and sold them at 60 ^ a box. How much did they get for both boxes? 3. If there are 24 boys and 32 girls in the school, how many children are there in the school ? 4. Six boys bought a dozen bananas and shared them equally. How many bananas did each boy get? 5. A farmer had 60 sheep. How many did he have after selling 20 sheep? 6. It is 2 miles from Arthur's home to his aunt's. On Saturday Arthur made two trips on his bicycle to his aunt's and return. How many miles did he ride ? 7. If it takes 3 yards of cloth to make one apron, how many aprons can be made from 12 yards ? a If oranges sell at 20^ a dozen, how many dozen can be bought for 60^ ? 9. What is the cost of 3 pounds of coffee at 30^ a pound? 10. What is the cost of five 2-cent stamps? 11. At 60 cents a yard, how much will | yd. of cloth cost? 12. How many gallons are 'there in 12 quarts? WRITTEN PROBLEMS 111 145. Written Exercises. a b C d e / (1 h 474 938 546 218 917 645 539 462 907 794 477 827 129 387 958 547 836 672 259 388 451 719 867 243 593 866 998 916 679 693 926 589 688 978 738 193 832 568 775 362 966 489 999 778 178 686 944 318 389 749 823 143 819 617 233 759 146. Written Problems. 1. Harry is saving his money to buy a bicycle that will cost $ 45. He has saved $ 38. How much more must he save before he can pay for the bicycle ? 2. Four boys went fishing. They paid 25^ for the use of a boat, 15^ for bait, 35^ for some lines, and 45^ for lunch. Find the whole cost of the trip. Find each boy's share of the expenses. 3. A man bought a horse for $95. For what must he sell the horse to gain $25 ? jiv. A boy had 60 marbles and sold one third of them. How many marbles did he sell ? How many had he left ? 5. A farmer sold three cows for the following sums : $ 28, $ 36, and $ 40. How much did he get for them ? 6. A farmer sold 4 cows at $32 each. How much did he get for them ? il2 ADDITIOxV AND SUBTRACTION SURFACES ABC D 147. 1. The surface of Fig. A is — of the surface of Fig. B. 2. The surface of Fig. ^ is — of the surface of Fig. C. 3. The surface of Fig. B is — of the surface of Fig. a 4. The surface of Fig. ^ is — of the surface of Fig. D. 5. The surface of Fig. ^ is — of the surface of Fig. D, 6. The surface of Fig. ^ is — times the surface of Fig. A. 7. The surface of Fig. D is four times the sur- face of Fig. — . 8. The surface of Fig. D is two times the surface of Fig. — . 9. The ratio of Fig. A to Fig. 5 is — . 10. The ratio of Fig. C to Fig. A is — . 11. What part of the surface of Fig. C is equal to the surface of Fig. A ? 12. Three times Fig. A is equal to Fig. — . 13. If Fig. A represents 2 square inches, Fig. B will represent — square inches. LENGTH 113 148. 1. How long is this book ? 2. The unit of length used to measure short dis- tances is the inch. 3. How long is this room ? The foot is the imit of measure next in length to the inch. We use the unit 1 foot in measuring the length of a room. 4. What is the unit of length in measuring cloth ? 5. The rod and the mile are each units of length. These are used in measuring long distances. 6. Study the inch, the foot, and the yard. See Lesson VII, p. 21. 7. Draw a line 12 inches long. Divide the line into inches. The unit of measure of the line is — . 8. Draw a line 3 feet long. 9. Three feet are 1 yard. 10. One foot is — third of a yard. 11. In 5 feet there is — yard and — feet. 12. What part of one foot is one inch ? 13. Six inches are — twelfths of 12 inches. 14. Divide 12 inches into 3 inches. 15. In 12 inches there are — 3 inches. 16. Divide 12 inches into 4 inches. 17. In 12 inches there are — 4 inches. 18. Memorize : Twelve inches are one foot, Three feet are one yard. 1st P.k Ahitii— S ^ 114 ADDITION AND SUBTRACTION 149. 1. Your desk top has length. Has it width ? 2. Anything that has length and width has area. 3. Have the sides of this room area ? 4. Has a book cover area ? 5. A square inch is a square whose sides are each one inch. Draw a square inch. 6. A square inch is the smallest unit of area. 7. Name a unit of area larger than the square inch. a Draw an oblong 3 in. long and 1 in. wide. Divide the oblong into square inches. How many square inches are there in the oblong? 9. Draw an oblong 3 in. long and 2 in. wade. Divide it into square inches. How many square inches are there in the oblong ? What is the area of the oblong? How many 3 sq. in. are there in the oblong ? 10. In Problem 9, two square inches are what part of the oblong ? Three square inches are what part of the oblong ? 11. Draw an oblong 4 in. long and 3 in. wide. How many 4 sq. in. can be made of the oblong ? What is the area of the oblong ? 12. Draw an oblong 4 in. long and 2 in. wide. How many 4 sq. in. can be made of this oblong? What is the area of the oblong ? 13. Draw an oblong 4 in. long and wide enough to contain 4 sq. in. What is its area ? CHAPTER IV MULTIPLICATION AND DIVISION SIMPLE FRACTIONS, COMPOUND NUMBERS, REVIEWS MULTIPLICATION — LESSON 150. Oral Problems.* 1. How much will 2 chairs cost at $ 4 each ? Model for oral recitation : Since 1 chair costs $ 4, 2 chairs will cost 2 times $ 4, or $ 8. Model for written recitation : $ 4, cost of 1 chair. x2 $ 8, cost of 2 chairs. 2. How much will 2 clocks cost at $ 6 each ? 3. At $ 3 a pair, how much will 2 pairs of shoes cost? 4. How much will 2 tables cost at $ 5 each ? 5. At 6^ each, how much will 2 oranges cost ? 6. How much will 3 hats cost at $ 4 each ? 7. There are 4 quarts in a gallon. How many quarts are there in 2 gallons ? * Drill should be given upon these and similar problems until the pupils are familiar with the language forms used in the analysis. The written form should be taken up after the oral form has been mastered. Apply these forms to similar problems on the succeeding pages of the text. 115 116 MULTIPLICATION AND DIVISION 151. Oral Problems.* 1. If 2 chairs cost $ 6, what is the cost of 1 chair ? Model for oral recitation: If 2 chairs cost $6, 1 chair will cost one half of $6, or $3. Model for written recitation : $ 3, cost of 1 chair. 2)$G, cost of 2 chairs. 2. If 2 stoves cost $ 10, what is the cost of 1 stove ? 3. If 2 tables cost $ 8, what is the cost of 1 table ? 4. If 2 tablets cost 12^, what is the cost of 1 tablet ? 5. If 3 boxes of berries cost 15^, what is the cost of 1 box ? 6. If 3 pencils cost 6^, what is the cost of 1 pencil ? 7. If 3 hats cost $ 9, what is the cost of 1 hat ? 8. If 4 pairs of shoes cost $12, what is the cost of 1 pair? 9. If 1 yd. of cloth costs 12^, what is the cost of iyd.? . 10. If 1 yd. of ribbon costs 8^, what is the cost of iyard? 11. What is the cost of i yd. of cloth at 15^ a yard ? 12. What is the cost of h doz. eggs at 12^ a dozen ? 13. If 4 chairs cost $ 12, what is the cost of 1 chair ? 14. What is the cost of 1 stove at the rate of 2 stoves for $12? * See note, p. 115. MULTIPLICATION — LESSON C 117 152. Oral Exercises. 1. Add a coliunn of four 4's ; of five 3's. 2. Count by 4's to 16 ; by 3's to 15. 3. Add a column of seven 2's ; of eight 2's. 4. Count by 2's to 14 ; by 2's to 16. 5. Four 4's are — ; five 3's are — ; three 5's are — . 6. Seven 2's are — ; two 7's are — ; eight 2's are — . 7. In 16 there are — 4's. There are — 2's in 16. 8. How many nickels are there in 15 cents? 9. Memorize : 4 8 7 2 2 x4 x2 x2 x8 x7 Te 16 14 16 14 10. 7 is read two 7's are 14. It may also be X 2 read, two times 7 is 14. 11. A boy bought 3 oranges at 5 )^ each. How much did he pay for all ? 12. At 3 ^ each, how much will 5 pencils cost ? 13. Frank has $4 and Arthur has four times as much money. How much money has Arthur? 14. At 1 2 each, how much will 8 hats cost? 15. There are 7 days in one week. How many days are there in 2 weeks? 16. Etiiel worked 8 problems and Edna worked twice as many. How many problems did Edna work ? 118 MULTIPLICATION AND DIVISION 153. Study Exercises. 4 8 7 2 2 x4 x2 x2 ,x8 x7 Study the above exercises until you can give the products without hesitation. Give the products from right to left, adding 3 to each product, thus: 14, 17; 16, 19 ; etc. 154. Written Exercises. abed e • / 9 1. 2034 3140 4213 1324 4321 8400 2341 x4 x4 x4 x4 x4 x4 x4 2. 3023 3333 2323 2030 1302 3032 3021 x5 x5 x5 x5 x5 x5 x5 3. 1212 2021 2222 2012 1220 2222 2121 x6 x6 x7 x7 x7 x8 x8 Multiply 457 by 20. Model : 457 x20 9140 a b 3457 5678 x20 x20 times 457 is 0. Write under the in the multiplier. 2 tunes 7 is 14. Write the 4 under the 2 in the multiplier, and continue. c d e f q 6785 3467 4576 6587 3478 x20 x20 x20 x20 x 20 6. 3425 x30 5243 4035 2304 4230 3040 2130 x30 x30 x40 x40 x 40 x 50 ORAL EXERCISES 119 155. Oral Exercises. Add each column as indicated in Step C, p. 49. a b c d e / 9 A ^ J k I TO 7 8 8 7 9 8 7 6 9 8 7 4 2 3 7 2 3 1 2 3 4 1 2 3 6 8 6 5 8 8 4 4 5 5 3 8 7 4 2 4 8 1 2 6 6 5 5 7 2 5 3 8 7 7 8 7 9 8 7 6 9 6 5 7 2 3 5 2 3 1 2 3 4 1 4 3 6 8 6 5 8 8 4 8 5 5 3 6 5 3 8 7 8 9 7 9 4 .7 6 9 8 6 5 4 156. 1. Show by objects how many 2 books there are in 3 books ; in 5 books ; in 7 books. 2. Show how many 3 boys there are in 7 boys. 3. In 7 there are — 2's and — remainder. 4. How many 2's are there in 70 ? Model : 35 In 7 there are three 2's and 1 2)70. remainder. Write 3 above the 7, and think the 1 before 0. In 10 there are 5 twos. Write 5 above the 0. There are 35 twos in 70. One half of 70 is — . 157. Divide: a b c d e / 1- 2)70 2)50 2)92 2)76 2)34 3)42 2. 3)102 3)105 3)300 3)420 3)720 3)672. 3. 2)530 2)302 2)710 2)930 3)1032 3)7032. 120 MULTII'LICATION AND DIVISION DIVISION — LESSON C 158. .1. In $4 there are — $2. In $5 there are — $2 and $ — reniainder. 2. What is 1 of 5 ? What is l of 11 ? 3. To find one half of a number, divide it by — . 4. Find i of 9. Model: 4^ There are four 2's in 9, and 2)9. 1 remainder. The remainder is written over the divisor as above. 5. One half of 9 is — . Nine divided by 2 is — . 6. Find 1 of 7 ; of 11 ; of 13 ; of 5 ; of 10. 7. Find 1 of 4 ; of 7 ; of 10 ; of 5 ; of 8. a Find J- of 5 ; of 6 ; of 7 ; of 8 ; of 9 ; of 10. 9. Six divided by 2 may be written : 2)6, or 6 -i- 2, orf. 159. Bead and memorize : 4 2 4)16 8)16 2 7)14 7 2)14 8 2)16 160. Supply quotients in the following : 1. 6^2 = .r. 7. 10^2 = a:. 13. 15^-3 = 0;. 2. S-^i=X. a 12^6 = a:. 14. 12^2 = x. 3. 6^3 = a:. 9. 15^5 = x. 15. U-i-2 = x. 4. 9-5-3 = a;. 10. 16-5-2 = a:. 16. 16^8 = x'. 5. 8-^2-a:. 11. 12^3 = a:. 17. 14 -*- 7 = X. 6. 4-^2 = 0:. 12. 16H-4=ic. 18. 12-h4 = x. 161. Drill Exercises. 1. a 2. ^ 6 12 3 3 3. ^ 4 1_5 '5 5. JL^ 3 4 -12. 6 H. 1_2_ 4 2 \TIOX- -LESSON D 121 Give answers : C d e / 14 13 4 3 -2" -3- 3 2 15 14 5 5 3 3 3 2 16 10 7 5 "4" "3 3 4 16 11 8 6 -2' -3" 3 4 16 11 7 7 ~8- 2" 2 4 14 13 9 9 7 2 2 4 162. Subtract: a 6 c d e f 9 1. 8457 9745 9944 '4715 6486 9312 7458 2958 3887 3146 _859 2688 6317 2769 2. 9236 5211 8452 8294 9732 8290 7408 8267 2343 _674 2705 5846 1304 980 MULTIPLICATION — LESSON D 163. 1. Add a column of six 3's ; of nine 2's. 2. Two 9's are — ; nine 2's are — . 3. Find the sum of three 7's ; of seven 3's. 4. In 3 weeks there are — days. 164. Memorize: 6 9 7 3 2 • 3 x^ x^x3 x6 x9 x7 18 Ts li 18 18 2T 122 MULTIPLICATION AND DIVISION 165. Study Exercises. 6 9 7 3 2 3 x3 x2 x3 x6 x9 x7 Study the above exercises until you can give the products without hesitation. Give the products in the above exercises from right to left. Give the products from right to left, adding 3, 4, and 5 to each product, thus, adding 3 : 21, 24 ; 18, 21; etc. Give the products from right to left, adding 6, 7, and 8 to each product, thus, adding 6: 21, 27; 18, 24; etc. 166. Written Exercises. iviuitipiy : a b 1. 7897 9789 2 2 c 8789 2 6798 2 • e 7968 2 / 9687 2 2. 5467 3 7654 3 4567 3 7456 3 6745 3 5764 3 3. 4321 4 1423 4 3012 5 2130 5 2301 6 3210 6 4. 3023 .7 2220 8 2323 7 3020 6 2343 4 3223 5 5. 7605 20 6750 30 4032 40 3120 60 3213 50 1203 50 DIVISION — LESSON D 128 DIVISION — LESSON D 167. 1. Memorize tlie following : 6 • _9 _7 _3 _2 _3 3p 2)T8 3)21 6)18 9)18 7)2l 2. A boy had 18 marbles and sold \ of them. How many marbles did he sell ? How many marbles did he have left ? 3. How many weeks are there in 21 days ? 4. A girl had 18 cents. She spent \ of her money for some paper. What was the cost of the paper ? How much money had she left ? 5. If 18 apples are divided equally among six boys, what part of the whole number of apples will each boy receive ? 6. If 18 pencils are divided into 2 equal groups, Jiow many pencils will there be in each group ? 7. What part of 18 inches are 6 inches ? a What is the ratio of 21 to 7 ? Of 7 to 21 ? 9. In finding the number of $3 there are in $21, the unit of measure is — , and the quantity measured is — . 10. What part of 18 feet are 2 feet ? Are 9 feet ? 11. What is the ratio of 2 feet to 18 feet ? 12. Find! of $21. Find 1 of $18. 13. How many 2-cent stamps can be bought for 18 cents ? 124 MULTIPLTCATIOX AND DIVISION 168. Study Exercises. 3)l8 2)"T8 3)2l 6)18" 9)18' 7)2T 3)T9 2)19 3)22 6)19 9)l9 7)22 3)"20 2)17 3)"23 6)l3 5)l6 7)23 3)16 3)17 8)17 4)17 7)15 7)l6 In the study of the above exercises use the follow- ing models : (a) In 19 there are six 3's and 1 remainder. (h) One third of 19 is 6J. Study the above exercises until you can give the quotients without hesitation. 169. Written Exercises. a 6 c d e 1. 2)1980 2)1330 2)9398 2)5112 2)1776 2. 2)1816 2)1412 2)1306 2)3170 2)1360' 3. 3)1005 3)1215 3)1998 3)1665 3)2322 4. 3)2001 ' 3)1710 3)1671 3)1356 3)5109 S. 4)1768 4)1372 4)1736 4)1216 4)9736 6. 5)1160 5)1615 5)1510 5)1155 5)1665 7. 6)1812 6)1998 6)1332 6)7206 6)7938 8. 7)2331 7)1631 7)9184 7)1561 7)7140 9. 8)1776 8)9768 8)1608 8)1760 8)1696 10. 9)1809 9)1998 9)1098 9)1089 9)1908 DIVISIOX — LESSON D 125 170. Oral Problems.* 1. At $2 each, how many chairs can be bought for $8? Model for oral recitation : Since 1 chair costs $ 2, as many chairs can be bought for $8 as there are $2 in $8, or 4. Four chairs can be bought for $8. Model for written recitation : _4 chairs for $8. cost of 1 chair, $2)$ 8 2. If 1 pair of shoes costs $ 3, how many pairs can be bought for $12? 3. If 1 box of berries costs 5,^, how many boxes can be bought for 20^? 4. At $3 each, how many hats can be bought for $9? 5. How many yards of ribbon at 4^ a yard can be bought for 16^? 6. At 6^ each, how many tablets can be bought for 18^? 7. If a boy earns $ 4 a week, in how many weeks will he earn $20? 8. A farmer sold some sheep for $6 each. He received $18. How many sheep did he sell? 9. If 2 girls sit in each seat, how many seats will 20 girls occupy ? 10. A girl earns 8^ a day. In how many days will she earn 16^? * See note, p. 115. 126 MULTIPLICATION AND DIVISION 171. Written Problems. 1. A man bought 4 cows. For one he paid $27, for another $32, for another $36, for another $40. How much did he pay for the four cows ? 2. At $32 each, what will be the cost of 4 cows? 3. Can you find the cost of the cows in the first problem by multiplication ? 4. Can you find the cost of the cows in the second problem by addition ? 5. When can you use either addition or multipli- cation to find the cost of cows ? 6. When you can use either addition or multipli- cation, which is the better to use ? Why ? 7. A man bought a farm for $3675, and sold it for $ 5000. Did he gain or lose, and how many dollars ? 8. A man bought 6 sheep at $3 each and sold them for $ 25. Did he gain or lose, and how much ? 9. A farmer owned 1 860 acres of land. He divided his land into 3 farms, with the same number of acres in each. How many acres were there in each of the farms ? 10. At $4 each, how many sheep can be bought for $128? 11. At $ 5 each, how many barrels of flour can be bought for $65? 12. At 23^ a yard, what will be the cost of 6 yards of cloth ? ORAL PROBLEMS 127 172. Oral Problems. 1. How many halves are there in 1 apple ? In 1 circle ? In 1 dollar ? In 1 day ? 2. How many halves are there in 2 apples ? In 2i apples ? In 3 apples ? In o^ apples ? In 4 J days ? 3. How many apples must be cut into halves in order to get | apples ? | apples ? | apples ? 4. Tell how many whole apples each of the fol- lowing is equal to : |- apples, |- apples, | apples. 5. A man paid 4 boys | dollar each for helping him on a Saturday. How many dollars did he pay them all ? 6. There were 8 children at a party. A lady gave one half of an orange to each. How many oranges did she give to all ? 7. 2^ apples = f apples ; |- apples = — J apples ; 3^ apples = f apples. 8. 4i days = | days ; 5 J dollars = | dollars. 9. 6 1 feet = I feet ; 1 J inches = f inches. 10. f apples = — apples ; ^ oranges = — i oranges. 11. Tell how many half apples each of the follow- ing is equal to : 2 apples, 2^ apples, 4 apples, 41 apples, 5^ apples, 31 apples, 6 apples. 12. Seven half dollars are equal to how many dollars ? 13. A girl spent 5 half days in the city. How much more than 2 days did she spend there ? 128 MULTIPLICATION AND DIVISION 173. Drill Exercises. G X 2 is read, 6 multiplied hy 2. It is the same as t/ Supply products for x, and add to each product the number above the column as in a : 12, 14 ; IG, 18. Give answers only : (2) (3) ■ (5) (7) a b erf 1. Gx2 = 12,14 8x2 = 16,19 7x3 = a: 3x2 = a: 2. 4x4 = lG,18 4x3 = x 2x5 = x oxS = x 3. 5x2 = x 6x3 = a: 4x2 = x 3x4 = a; 4. 3x3 = a; 7x2 = x 3x5 = a: 2x6 = x 5. 2x3 = a; Sx(j = x 9x2-a; 2x8 = a; 174. Drill Exercises. Supply quotients in place of x. a 1. 18^3 = a: b 12H-4 = a; c 16^2 = aj d 19-^3 = a: 2. 14-^7 = 0: lS-^G = x 18-4-9 = a: 11^2 = a; 3. 21^3 = aj 16^8 = ^ 15-^3 = x 13H-6 = a; 175. Written Exercises. a b c 1. 978,978 897,897 679, x2 x2 d 769 587,857 x2 x2 e 459,596 x2 2. 765,765 657,657 547,574 637,673 564,767 x3 x3 x3 x3 x3 3. 324,243 432,432 304,203 321,321 123,123 x4 x4 x4 x5 x6 ORAL EXERCISES 129 176. Oral Exercises. Add each column as indicated in Step C, p. 49. a b c d e / 9 h I i /t I m 3 9 8 7 6 G 1 7 6 9 9 5 7 5 3 2 4 1 3 9 3 4 1 3 5 6 5 7 8 6 9 t 8 7 1 4 6 1 9 7 5 9 3 8 4 2 3 9 6 4 4 3 5 3 2 A 1 5 8 6 8 7 8 6 7 5 7 8 6 9 5 4 4 2 3 2 8 3 4 8 9 7 8 9 9 8 2 2 7 9 7 8 6 7 8 9 6 6 8 9 8 6 9 8 Name five combinations whose smns are ten. When these combinations occur in a column, they may be taken together. Exercise a above may be added : 12, 22, 29, 39, 42. Add the above exercises in a similar manner. 177. Written Exercises. Sul^tract : a h 16,043 72,345 9,876 34,567 c 90,234 27,840 d 45,803 14,769 e 84,087 28,309 178. Oral Exercises. Add a column of ten 3's ; of five 4's ; of five 5's. Memorize : 8 9 5 3 3 4 5 x3 x3 x4 x8 x9 x5 x5 ~24 17 "20 "24 "27 "20 "25 IsT liK Anrni — MULTIPLICATION AND DIVISION 179. 8 x3 MULTIPLICATION — LESSON E Study Exercises. 9 x3 x4 3 x8 3 x9 4 x 5 X o Study the above exercises until you can give the products without hesitation. Give the products, from right to left ; adding 4 to each product ; adding 6 to each product. M^ 180 180. Multiply 3457 by 23. DEL : 3457 Multiply 3457 first by 3, and 23 write the product. Next, multiply 10371 by 2. Two 7's are 14. Write the 6914 4 under the 2. After completing 79511 the multiplication by 2, draw a line and add the products. 181 Written Exercisei Multiply : 1. 3457x23 2. 6789x23 3. 9876x32 4. 5647x32 5,^8975x23 6. 3240x35 7. 6978x32 8. 4036x23 9. 9380x32 10. 3243x45 11. 4545 X 54 12. 3454x45 13. 2050x54 14. 3524x54 15. 2320x67 16. 3023x67 17. 1323x67 18. 2032x67 19. 3223x89 20. 2130x98 21. 3231x89 22. 3020x98 23. 3123x84 24. 2323x29 25. 4534x15 26. 3750x13 27. 4567x20 ORAL riiOBLEMS 131 182. Oral Problems. 1. At $3 each, how many chairs can be bought for $18? 2. How much will 4 tables cost at $ 5 each ? 3. If 8 hats cost $16, how much w411 1 hat cost? 4. At $ 4 each, how many desks can be bought for $12? 5. How much will 6 tablets cost at 3 ^ each ? 6. How many pencils at 2^ each can be bought for 8^? :h 7. At the rate of 4 for 12^, what is the cost of 1 pencil ? 8. If 3 packages of popcorn cost 15y, what is the cost of 1 package ? 9. How much will 3 boxes of berries cost at 6y per box? 10. Four boys rode on a street car. They paid 5^ each. How much did it cost the four boys ? 11. Mary spent 3 weeks with her aunt. How many days did she spend with her ? 12. George paid 21^ for 3 tablets. How much was this for each tablet ? 13. Ethel walks 2 miles to school. How far does she walk in going and coming each day ? How far does she walk in 1 week ? 14. Frank is 7 years old. His brother is 21 years old. His brother is how many times as old as Frank ? 132 MIILTIPLICATIOX AND DIVISION 183. Written Problems. 1. A boy I'ode 15 miles in 3 hours. What was the average rate per hour ? 2. A train runs 96 miles in 4 hours. What is the average rate per hour ? 3. A grocer sold $ 1380 worth of goods in 6 days. Find the average daily sales for the week. 4. George weighs 84 lb. ; Walter weighs 76 lb. How much will the two boys together weigh ? What is the average weight of the two boys ? Which of the boys weighs more than the average weight ? 5. There are 50 pupils in Room A ; 43 pupils in Room B; 40 pupils in Room C; and 39 pupils in Room D. How many pupils are there in the four rooms ? Find the average number of pupils in a room. Which of the rooms have more than the average number of pupils? Which have less than the average number ? Has any room the average number of pupils ? 6. A farmer sold a grocer 7 lb. of butter at 23 ^ a pound. What was the value of the butter? The farmer bought of the grocer 2 lb. of coffee at 25 ^ a pound, and 3 lb. of tea at 35^ a pound. What was the value of the coffee and tea ? Did the farmer owe the grocer or the grocer owe the farmer, and how much ? 7. Find i of $1760. Find ^ of $1600. 8. Which is the more, i of $ 120 or | of $ 176 ? DIVISION — LESSON E 133 DIVISION — LESSON E 184. 1. Memorize tlte following : 8 9 5 3 3 4 5 3)24 3)27 4)20 8)24 9)27 5)20 5)25 2. Eight is ^ of — . 5 is i of — . 3 is 1 of 3. Three is l of — . 4 is i of — . 5 is 1^ of — 4. What is i of $ 24 ? What is i of $ 25 ? 5. What is i of 20? What is 1 of 27? 6. In 27 there are — 3's. In 28 there are — 3's, and — remainder. 7. In 24 there are — 3's. In 25 there are — 3's, and — remainder. 8. What is 1 of 20? What is 1 of 21? 9. What is tlie ratio of 5 to 20 ? Of 20 to 5 ? 10. How many $ 5 units are there in $ 25 ? 11. How many $1 imits are there in $25? 12. How many units of 3 feet are there in 24 feet ? 13. How many times must the yardstick be applied in measuring 24 feet? 14. Nine feet are i of — . $4 are ^ of $ — . 15. There are 24 hours in one day. How many hours are there in l of a day ? 16. A boy had 20^. He bought a tablet that cost \ of his money. W^hat was the cost of the tablet ? The boy had — ^ left. 17. What is 1 of 20^ ? What is 1 of 20^ ? 134 MULTIPLICATION AND DIVISION 185. Study Exercises. 3)24 3)27 4)20 . 8)"24 9)27 5)"20 5)"25 3)25 3)"28 4)"21 8)25 9)18 5)'2T 5)26 3)26 3)29 4)22 8)26 9)29 5)"22 5)27 Study the above exercises until you can give the quotients without hesitation. Give the quotients and the remainders, if any. Give the quotients with the remainders expressed as fractions. Review Exercise 168, p. 124. 186. Written Exercises. Use the numbers above the columns as divisors : q 9 4, 3 5 6 1. 2^2,724. 7. 201,620 13. 202,520 19. 181,213 2. 272,427 8. 177,808 14. 252,025 20. 139,392 3. 251,515 9. 217,371 15. 152,520 21. 738,798 4. 266,664 10. 222,200 16. 101,520 22. 193,278 5. 267,267 11. 982,208 17. 252,015 23. 792,192 6. 287,878 12. 140,142 18. 267,676 24. 180,192 187. Write in a column and add : 1. $1045, $72.05, $108.75, $9.18, $.75, $704. 2. $304.50, $40.20, $1000, $.85, $19.90, $1.25. 3. $6.40, $200.45, $3.05, $89.26, $6, $600, $8.30. 4. $300, $8, $4000, $.12, $20, $10.50, $.64. DRILL EXERCISES 135 188. Drill Exercises. Give quotients and express the remainders as fractions : 1 _2Jl 2_1 _2_2 c 2_0 _2_1 _2_2 -.^ 2_7_ 2_8 2 9 J- 4 ? 4 ? 4-- o- 5 9 "5 ? 5 • •'"*•• 9 ? 9 ? 9 • o 1_5_ 1_6 1_1 „ 1_6 1_7_ 18. TO 10 17 1_8 ^' 35353* '• 45454- ■'■^- 8"? 8"5 "8 • o 2_5 2_6 2_7 a JL8_ 1_9. 2_0_ to 2 1 2 2 2 3 •*• 5 5 5'5 "5 • **• 3 5 "3 ? 3 • J-^- -7 5 7^5 ~7 • A 1-8 li> _2_Q. Q 12 IJl 14 lA 1« li) 2_0 *• 9 5 9"? 9 • ^- 6 5 '6 5 6* •'■*• '6 5 6"5 '6 ' c 2Jb 2_5 2_6 in JL4 15 16 ,« 15 16 17 S- "8'"5 85 -8-- 10. -S7-, -y-, -y-. 15. -J-, -5--, -5-. 189. Written Exercises. 1. Multiply $6.50 by 3. Model : $ 6.50 Multiply as in previous exer- 3 cises, and point off two places $ 19.50 for cents. 2. $656.50x2 7. $897.68x2 12. $302.23x7 3. $329.40x3 8. $950.75x3 13. $310.32x8 4. $345.54x4 9. $432.50x4 i4. $231.12x9 5. $453.45x5 10. $345.24x5 15. $330.30x9 6. $323.10x6 11. $230.20x6 16. $103.23x8 190. Written Problems. Give the analysis for each : 1. At $4.75 eachj what will be the cost of 3 sheep ? 2. What will be the cost of 4 chairs at $3.25 each? 3. At $42.50 each, what will be the cost of 4 cows? 4. A boy earns $15.75 a month. How much will he earn in 3 months ? 136 MULTIPLICATION AND DIVISION 5. How much will 3 railroad tickets from Chicago to San Francisco cost at $62.50 each? 6. A man pays $ 13.50 a month rent for a house. How much will this amount to in 4 months? 7. Find the cost of 3 tons of coal at $6.75 a ton. 8. A man bought sheep at $4 each. He paid $176 for the sheep. How many sheep did lie buy? 9. What is the unit of measure in Problem 8 ? 10. A man owned 360 acres of land. He sold ^ of it. How many acres did he sell? How many acres did he have left ? 191. Oral Exercises. Add as indicated in Step C, p. 49. a 6 C d e X 9 h i J Z: I m n 2 4 2 3 3 3 3 9 2 4 2 9 1 5 3 5 2 4 3 r 6 5 4 4 7 1 3 4 5 8 7 5 4 2 1 3 4 2 1 4 6 8 2 3 9 8 3 3 3 2 2 4 2 5 1 3 3 4 2 3 3 5 6 5 4 4 7 1 3 5 8 5 2 4 5 8 4 3 7 2 6 4 8 4 3 3 4 8 9 4 8 8 6 9 8 9 6 7 !) 7 6 2 6 8 8 9 7 9 6 1 6 3 Whenever two numbers whose sum is not more than 9 are to be added to 10, 20, 30, etc., take both numbers together. Exercise a above may be added: 12, 20, 25, 30, 35. Add the above exercises in a similar manner. MULTIPLICATION — LESSON F 137 MULTIPLICATION — LESSON F 192. 1. Memorize the following : 6 7 6 4 4 5 6 x4 x4 x5 x7 x6 x6 x6 "24 "28 30 28 24 30 36 2. Count by 5's to 30 ; by 6's to 36. 3. Count by 4's to 28 ; by 7's to 28. 4. There are 7 days in one week. How many days are there in 4 weeks ? 5. There are 4 quarts in one gallon. How many quarts are there in six gallons ? 6. What is the product of 4 and 7 ? Of 4 and 6 ? 7. Alice bought 6 pencils at 4^ each. She handed the clerk a 25-cent piece. How much change should she receive ? 8. A boy sold 6 papers at 5^ each. How much money did he receive for all ? 9. At $4 each, what will be the cost of 7 chairs ? 10. What will be the cost of 6 tablets at 6^ each? 11. At the rate of 5 marbles for a cent, how many marbles can be bought for 6 ^ ? 12. A girl bought 4 yd. of ribbon at 6^ a yard. She gave the clerk 25 cents. How much change should she receive ? 13. How many cents are 6 nickels ? How many dollars are six 5-dollar gold pieces ? 138 MULTIPLICATION AND DIVISION 193. Study Exercises. 6 7 6 4 4 5 6 x4 x4 x5 x7 x6 x6 x6 Study the above exercises until you can give the products without hesitation. Give the products from right to left, adding the following numbers to each product : 3, 4, 7, 8. .194. Multiply: 1. 677,676 X 34 ii. 654,564 x 56 21. 321,213 x 78 2. 765,756 X 24 12. 365,456 x 56 22. 203,320 x 68 3. 456,746 X 14 13. 246,365 x 65 23. 332,223 x 58 4. 375,647 X 34 i4. 654,321 x 56 24. 123,123 x 48 5. 263,746 X 24 15. 506,430 x 46 25. 301,203 x 38 6. 565,656 x 54 le. 434,343 x 67 26. 332,233 x 89 7. 456,546 X 45 17. 342,434 x 57 27. 312,013 x 79 8. 346,543 X 54 la 234,342 x 47 28. 120,320 x 69 9. 425,636 X 35 19. 324,130 x 37 29. 231,032 x 59 10. 654,321 X 45 20. 123,432 x 27 30. 123,320 x 49 195. Solve: $913.78 $935.36 $8312.75 $835.00 -$535.79 -$145.68 -$2353.76 -$135.25 What is the answer in division called ? DIVISION — LESSON F 139 DIVISION — LESSON F 196. 1. Memorize the folloiving : 6 '^ _5 _i _i __^ _^ 4)24 4)28 5)30 7)28 6)24 6)30 6)36 2. How many weeks are there in 28 days ? 3. How many gallons are there in 24 quarts ? 4. How many nickels are there in 30 cents ? 5. What is J of $24? What is i of $36 ? 6. $6isiof$— . $7isiof$— . 7. At 5^ each, how many oranges can be bought for 30^? 8. At $4 each, how many chairs can be bought for $28? 9. A girl had 30 cents. She spent ^ of her money for a tablet. What was the cost of the tablet ? 10. How many pounds of sugar at 6 ^ a pound will cost 24^? 11. What is the unit of measure in Problem 8 ? 12. There are 36 pupils in a schoolroom. There are 6 pupils seated in each row. How many rows of seats are there in the room ? 13. If each stove costs $ 7, how many stoves can be bought for $28? 14. If a boy earns $ 6 each month, how long will it take him to earn $24 ? 15. What is the ratio of 6 to 24 ? Of 24 to 6 ? 140 MULTIPLICATION AND DIVISION 197. Study Exercises. 4)24 4)28 5)30 7)28 6)24 6)30 6)36 4)25 4)29 5)3T 7)29 6)2'5 6)3"l 6)37 4)26 4)30 5)32 7)30 6)"26 6)32 6)38 4)27 4)31 5)33 7)"3l 7)27 6)T3 6)39 Study the above exercises until you can give the quotients without hesitation. Give the quotients and the remainders, if any. Give the quotients with the remainders expressed as fractions. 198. Written Exercises. Divide the numbers in each cohimn by the numbers above the columns. 4,3 5,2 6,3 7,2 15-242,824 11. 303,305 21. 243,024 31. 212,821 2A268,264 12. 252,030 22. 254,544 32. 296,968 35 264,268 4.''806,704 s.^307,048 13. 318,180 23. 266,736 33. 233,814 14. 323,230 24. 393,936 34. 226,257 15. 272,780 25. ■ 272,780 35. 309,401 6. 257,770 16. 267,605 26. 267,606 36. 156,268 7. 226,570 17. 813,215 27. 813,215 37. 870,401 a. 270,264 18. 627,110 28. 627,120 38. 994,714 9. 936,536 19. 758,230 29. 758,130 39. 169,547 10. 623,012 20. 126,315 30. 126,315 40. 714,931 DRILL EXERCISES 141 199. Drill Exercises. Give quotients with the remainders expressed as fractions : 1- ¥, ¥. ¥• 7. ¥. ¥. ¥• 13. ¥, ¥> ¥• 2. ¥, -¥-> -¥■ 8. ¥. ¥. ¥- 14. ¥. ¥. ¥• 3. V, ¥. ¥• 9. ¥. ¥, V- 15. ¥, ¥, 18 4 • 4- ¥' ¥> ¥• 10. ¥> ¥. ¥• 16. ¥- ¥> ¥• 5- ¥. ¥. -V- 11. ¥. ¥> ¥• ' 17. ¥, 1 7 ¥• 6. -¥ ¥. ¥- 12. ¥, ¥> ¥• 18. ¥> ¥, ¥■ 200. Oral Exercises. Add each cohnnii as indicated in Step C, p. 49. a b C d e / i^345,643 x 89 5. 759,857x56 i4. 876,543x67 23.5'654,321 x 89 6. 847,498 X 56 15. 345,678 x 67 ' 24.^123,456 x 89 7. 938,739 X 56 le. 235,786 x 67 25./605,640 x 89 8. 782,728x56 17. 768,547x67 26/546,365x89 9.. 975,985 X 56 la 647,835 x 67 27.^ 435,620 x 89 228. Written Exercises. 1. Multiply Exercises 1-9 above by 34. 2. Multiply Exercises 10-18 above by 23. 3. Multiply Exercises 19-27 above by 65. 229. Solve: a . b c d e $924.37 $810.35 $735.41 $806.31 $848.12 - $235.49 - $316.58 - $236.46 - $216.73 - $250.27 DTVISTON — LESSON I -161 DIVISION — LESSON I 230. 1. Memorize the following : '■ _8 _^ _8 _6 _6 __7 6)48 6)54 7)56 8)48 9)54 8)56 2. What part of 48 is 6 ? What part of 48 is 8 ? 3. Eight is what part of 56 ? Six is what part of 54? 4. What is the ratio of 6 to 54 ? Of 54 to 6 ? 5. How many times is the unit $8 contained in the quantity $48? 6. If $48 is divided into 8 equal amounts, how many dollars will there be in each part ? 7. If $9 is the unit that represents the cost of 1 table and $54 is the quantity that represents the cost of the tables bought, how many tables were bought ? 8. A girl spent 56 days with her aunt. How many weeks did she spend with her ? 9. A girl spent 48^ for lace that cost 8j^ a yard. How many yards did she buy ? 10. A box containing 6 lb. of raisins was bought for 54^. How much did the raisins cost per pound? 11. Eight pounds of sugar were bought for 48^. What was the cost of the sugar per pound ? 12. A boy spent 56^ in 7 weeks. What was the average amount spent each week ? 1st Rk Aimtii — 11 162 MULTIPLICATION AND DIVISION 231. Study Exercises. 6)48 6)54 7)56 8)l8 9)54 8)~56 6)49 6)55 7)57 8)49 9)55 8)57 6)50 6)56 7)58 8)50 9)56 8)58 6)51 6)57 7)59 8)51 9)57 8)59 6)52 6)58 7)60 8)52 9)58 8)60 6)53 6)59 7)61 8)53 9)59 8)61 Study the above exercises until you can give the quotients without hesitation. Give the quotients and the remainders, if any. Review Exercise 218, p. 154. 232. Written Exercises. Use the numbers above the columns as divisors : 6,5,4 7,6,4 8,5,3 9,2,4 a 6 c d 1. 484,236 564,942 484,032 544,536 2. 460,830 212,835 449,296 598,887 3. 496,968 549,927 493,328 580,158 4. 508,148 619,766 451,392 482,904 5. 533,262 607,936 373,248 273,834 6. 473,268 337,974 203,720 136,926 7. 496,428 198,695 267,544 122,007 8. 109,110 250,026 133,088 194,868 9. 995,316 117,936 195,696 220,950 10. 218,904 898,352 923,624 597,987 MULTIPLICATION — LESSON J 163 MULTIPLICATION — LESSON J 233. 1. Memorize the foi lloiving : 9 9 8 7 8 9 x7 63 x8 x8 72 64 x9 63 x9 72 x9 81 2. Count by 8's to 72 ; by 9's to 81. 3. Count by 7's to 63 ; by 6's to 54. 4. A furniture dealer sold 8 tables at $8 each. How much did he receive for them ? 5. What is the area of an oblong 9 inches long and 8 inches wide ? 6. What is the area of a flower bed 9 ft. long and 7 ft. wide ? 7. At 8^ a box, how much will 8 boxes of berries cost ? 8. What is the sum of nine 9's ? Of eight g's ? 9. There are 9 square feet in 1 square yard. How many square feet are there in 8 square yards ? 10. A girl bought 9 yards of cloth at 7^ a yard. She handed the clerk 75^. How much change should she receive ? 11. How many square inches are there in the sur- face of a piece of paper 8 in. wide and 9 in. long ? 12. Find the cost of 7 lb. of raisins at 9^ a pound. 13. What is the ratio of 8 to 72 ? Of 9 to 72 ? 164 MULTIPLICAXrON AND DIVISION 234. Study Exercises. 9 9 8 7 8 9 x7 x8 x8 x9 x9 x9 Study the above exercises until you can give the products without hesitation. Give the products from right to left, adding the following to each product : 6, 9, 7, 8. 235. Written Exercises. Multiply the numbers in each column by the follow- ing numbers : 89, 67, 45, 32, 30 : 1. a 987,798 b 543,435 c 859,437 d 163,530 2. 798,978 454,353 627,849 242,607 3. 679,896 345,345 790,486 387,652 4. 867,897 534,354 936,748 559,608 5. 967,898 435,435 382,197 772,002 6. 789,679 253,524 904,382 659,003 7. 986,789 425,342 678,452 870,004 8. 987,698 530,420 987,654 489,603 9. 896,789 836. Solve: 315,402 456,789 378,960 a 6 c d e 1. 85,123 96,317 94,164 86,543 74,239 -47,536 ■ -67,429 -57,369 -18,565 -24,571 2. 68,036 92,228 87,514 83,634 84,410 -40,068 -65,143 -28,436 -34,057 -47,565 DIVISION — LESSON J 165 DIVISION — LESSON J 237. 1. Memorize the folloioing : _9 9 _^ JT _8 __9 7)68 8)72" 8)64 9)63 9)72 9)81 2. What is \ of $ 63 ? What is i of $ 72 ? 3. At 8^ a box, how many boxes of berries can be bought for 64^? 4. A boy paid 63^ for 9 pounds of raisins. How much per pound was this ? 5. Nine boys paid for the lemonade for a class picnic. The cost of the lemonade was 81^. What was each boy's share of the expense ? 6. Eight girls gave a party. The expenses amounted to 72^. What was each girl's share of the expenses? 7. At $8 each, how many tables can be bought for $64? 8. What is the unit of measure in Problem 7 ? 9. Sixty-three trees were set out in 7 rows with the same number of trees in each row. How many trees were set in each row ? 10. An oblong containing 72 sq. in. is 9 in. long. How wide is it ? 11. How many pounds of candy at 9^ a pound can be bought for 72^? 166 MULTIPLICATION AND DIVISION 238. Study Exercises. 7)63 8)72 8)64 9)63 9)72 9)81 7)64 8)73 8)65 9)64 9)73 9)82 7)65 8)74 8)66 9)65 9)74 9)l53 7)66 8)75 8)67 9)66 9)75 9)84 Study the above until you can give the quotients without hesitation. Give the quotients and remainders, if any. Give the quotients with the remainders expressed as fractions. Increase the dividends in each of the above columns until each divisor is contained in the dividend 10 times. Review Exercises 218 and 231. 239. Written Exercises. Use the numbers above the columns as divisors : a b C d 7, 5 8, 6, 4 3, 9, 2 4, 5 1. 635,649 726,456 978,560 322,789 2. 213,542 324,048 384,271 263,867 3. 714,644 241,632 567,894 545,982 4. 286,573 738,768 306,752 756,789 5. 698,572 516,976 983,084 295,837 6. 678,573 206,344 542,301 106,967 7. 599,186 767,592 894,517 408,432 8. 608,033 769,896 675,214 702,002 ORAL PROBLEMS 167 240. Oral Problems.* 1. If 4 chairs cost $12, how many chairs can be bought for $21? Model for oral recitation : If 4 chairs cost $ 12, 1 chair will cost ^ of $12, or $3. If 1 chair costs $3, as many chairs can be bought for $21 as there are $3 in $21, or 7. 7 chairs can be bought for $21. Model for written recitation : $3, cost of 1 chair. 12, cost of 4 chairs. 7 chairs for ^21. cost of 1 chair, $3)$ 21 2. If 3 tables cost $18, how many tables can be bought for $30? 3. If 2 tons of coal cost $14, how many tons can be bought for $28? 4. How many yards of ribbon can be bought for 20^, if 3 yd. cost 12^? 5. How many tablets can be bought for 40^, if 3 tablets cost 24^? 6. At the rate of 3 boxes for 18^, how many boxes of berries can be bought for 36^ ? 7. How many pencils can be bought for 24^, if 3 pencils cost 9^? 8. If Alice uses 6 lemons to make 2 pies, how many pies will a dozen lemons make ? * See note, p. 115. 168 MULTIPLICATION AND DIVISION 241. Oral Problems. Give the analysis for each : 1. How much will 4 trunks cost at $ 7 each ? 2. A boy bought 36 ^ worth of sugar at G ^ a pound. How many pounds did he buy ? 3. A grocer sold 6 qt. of berries at 8^ a quart. How much did he receive for them ? 4. If 6 pairs of shoes cost $ 18, what is the cost of 4 pairs of shoes ? 5. How many desks can be bought for | 32, if 6 desks cost $ 24 ? 6. How far will a boy ride in 8 hours, if he travels at the rate of 12 miles in 3 hours ? 7. If 5 apples cost 10^, how much will 8 apples cost at the same rate ? 8. If 4 bunches of firecrackers cost 20^, how much will 6 bunches cost at the same rate ? 9. How many sacks of potatoes at $ 2 a sack will pay for 4 tons of coal at $ 6 a ton ? 10. If it takes a boy 12 minutes to ride f of a mile, how long, at the same rate, will it take him to ride a mile ? 11. If it takes 2 men 1 day to build a fence, in what time can 1 man build it ? 12. If an acre of land is worth $ 45, how much is I" of an acre worth at the same rate ? 13. A grocer sold |- of a box of apples for 40^. How much was the box worth at the same rate ? WRITTEX PROBLEMS 169 213. Written Problems. 1. A farmer had 360 acres of land. He sold f of it at $ 60 an acre. How much did he receive for it ? 2. A man sold f of his farm for $4200. What, at the same rate, was the value of the farm ? 3. A boy had $36 in a bank. He drew out f of it to pay for a bicj^cle. How much did the bicycle cost him ? 4. In an orchard containing 480 trees | of the trees are orange trees and the remainder are lemon trees. Find how many of each kind there are in the oTchard. 5. A farmer sold 3 cows for $ 45 each and 2 horses for $130 each. He deposited in a bank f of the money received. Find the amount of his deposit. 6. A man bought 8 horses at an average cost of $79. He sold them all for $760. How much did he make on them ? Find the average amount made on each horse. 7. A man bought 9 cows for $225 and sold them for $315. How much did he make on them? Find the average amount made on each cow. 8. A farmer sold 360 sacks of potatoes, which was I of his entire crop, at. $2 a sack. What was the value of his entire crop at the same rate ? 9. How many weeks must a boy work at $3 a week to pay for a suit of clothes that costs $ 12 ? 170 MULTIPLICATION AND DIVISION 243. Drill Exercises. Give quotients with the remainders expressed as fractions : 1. ¥> ¥> ¥• 7. ¥. ¥> ¥- 13. ¥> ¥, ¥- 2. ¥. ¥, ¥• a ¥, ¥. ¥• 14. ¥, ¥. ¥- 3. ¥-> ¥> ¥• 9. ¥> ¥- ¥• 15. ¥-.¥,¥• 4. ¥, ¥. ¥• lo. ¥> ¥' ¥• 16. \S ¥, ¥• 5. ¥> ¥, ¥• 11. ¥> ¥, ¥- 17. ¥. ¥, ¥• 6. ¥> -¥> ¥• 12. ¥v¥. -V- 18. ¥, ¥. ¥• 244. Oral Exercises. Add each column as indicated in Step C, p. 49. a 6 c d e / S' h i J fc ; m 6 8 3 6 8 3 8 . 2 8 3 4 4 3 3 4 2 2 4 6 4 3 7 9 2 8 9 2 5 3 2 5 8 9 4 6 7 3 5 7 6 8 9 C 8 3 5 5 5 2 7 4 5 3 6 2 2 4 6 6 6 4 3 2 2 2 2 4 3 2 5 4 8 7 3 6 5 3 4 6 5 7 7 3 3 3 8 2 4 3 8 8 8 8 9 8 7 2 7 9 1 7 4 9 3 Two addends whose sum is ten or less may be taken as one number. Exercise a above may be added : 14, 19, 27, 36. It may also be added : 16, 25, 30, 36. Add the above exercises in a similar manner. WRITTEN PROBLEMS 171 245. Written Problems. 1. A man bought 9 horses for $ 945 and sold them for $125 each. Find the amomit of his gain or loss. What was the average gain or loss on each horse ? 2. A grocer bought 6 barrels of apples at $3.25 per barrel and sold the*m all for $25. Find the amount of his gain or loss. 3. A farm of 189 acres was bought for $65 an acre and sold for $80 an acre. Find the gain per acre. Find the gain on the entire farm. 4. A hardware merchant sold stoves at $9 each. During the year he sold $405 worth of stoves. How many stoves did he sell ? His profit was $ 2 on each stove sold. How. much did he make on the sale of stoves during the year ? 5. A man sold |- of his farm for $3600. What part of his farm was left ? At the same rate, what was the vahie of the part that was left? What was the value of the entire farm ? 6. A harness cost $30. This was f of the cost of the buggy. Find the cost of the 1 i^y. The har- ness and buggy together cost ^ of ^Le cost of the horse. Find the cost of the horse. Find the cost of all three. 7. A man had 288 miles to travel. He rode f of the distance on his wheel and the remainder on a train. How far did he ride on each ? 172 MULTIPLICATION AND DIVISION MULTIPLICATION -LESSON K 246. Oral Exercises. 7 75 64 70 354 100 xlO xlO xlO xlO xlO X 10 70 750 640 700 3540 1000 1. Compare the products in the above exercises with the multiplicands. How do they diit'er ? 2. Can you give a short method of multiplying a number by 1 ? 3. Multiply each of the following numbers by 10 : 34, 45, 8, 90, 11, 12, 524, 670, 200. 4. Can you give a short method of multiplying a number by 1 00 ? 5. Multiply each of the numbers in Problem 3 Ijy 100. 6. Each of the following numbers is 10 times what number : 60, 80, 90, 950, 100, 300, 760, 7600 ? 7. Each of the following numbers is 100 times what number : 600, 100, 5000, 3700, 9500, 85,000 ? 8. Divide each of the following 'numbers' by 10: 80, 110, 210, 340, 450, 6050. 9. Give a short method of dividing a number that ends in zero by 10. 10. A number that does not end in zero may be divided by 10 thus: 85-?- 10 is 8.5. A point called a decimal point is placed before the right-hand figure of the number. This divides it by 10. The answer is read, e'uiht (Uid five tenths. It is the same as S^^''^. MULTIPLICATION — LESSON L 173 MULTIPLICATION - -LESSON L 247. Oral Exercises. 3 4 5 6 11 11 xll xll xll xll x4 x6 33 44 55 66 44 66 Study these exercises to find a short way of mul- tiplying any number from 1 to 9 by 11. Multiply by 11: 8, 7,4, 3, 1,2, 9, 6, 5. Multiply 11 by each of the above numbers. MULTIPLICATION — LESSON M 248. Oral Exercises. 1. The number 12 is used in many of our measure- ments. There are 12 inches in 1 foot ; there are 12 months in 1 year ; there are 12 things in 1 dozen ; and the clock face is divided into 12 parts. Twelve dozen things are sometimes put together and called a gross. Is there a gross of crayon in a full box ? 2. Memorize : 12 12 12 12 12 x3 x4 x5' x6 x7 36 • 48 60 7^ 84 12 12 12 12 12 x8 x9 xlO xll xl2 96 108 120 132 144 3. How many oranges are 5 doz. oranges ? 4. How many months are 9 years? 12 years? 174 MULTIPLICATION AND DIVISION 249. Table of Products and Quotients. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 15 16 20 20 24 28 32 36 40 44 48 5 10 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72' 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 LONG MEASURE* 250. Distance is measured in inches, feet, yards, rods, and miles. The yard is the standard unit of length. The other units are derived from it. 251. Menwjize : 12 inches (in.) = 1 foot (ft.) 3 feet = 1 yard (yd.) 5 1 yards, or 16 J feet= 1 rod (rd.) 320 rods = 1 mile (mi.) Imile =1760 yd. = 5280 ft. * Review Exercise 140, p. 114. SQUARE MEASURE 175 SQUARE MEASURE 252. 1. Using a yardstick, draAV on the blackboard a square whose side is one yard. This is a square yard. 2. Using a foot rule, draw a square whose side is one foot. What is this square called ? 3. Divide the square yard into square feet. How many square feet are there in one square yard ? 4. Draw a square inch. Divide a square foot into square inches. How many square inches are there in one square foot ? 12x12 = — . 5. Measure on the ground a square whose side is one rod. Drive a stake at each corner of it. This is called a — . 6. It takes 160 square rods to make one acre. A piece of land 16 rd. long and 10 rd. wide is one acre. A piece of land 20 rd. long and — rd. wide is one acre.''^ 7. A piece of land one mile square contains 640 acres. This is called a section. 253. Memorize : 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30^ square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) *If feasible, have the children measure off one acre on the school grounds or in an adjoining field. 176 Mri/riPLICATTON AND DIVISION 254. Written Problems. 1. Find the number of square feet in a walk 50 ft. long and 4 ft. wide. 2. How much will it co^t at $ .12 a square foot to lay a cement walk 50 ft. long and 5. ft. wide ? 3. Find the area of a walk 60 ft. long and 5 ft. wide. 4. How many square inches are there in o sq. ft.? In 31 sq. ft. ? 5. How many square inches are there in 3 sq. ft. and 32sq. in.? 6. Find the number of square feet there are on the floor of your schoolroom. 7. Find the number of square rods there are in your school yard. 8. Find the number of square yards of blackboard there are in your schoolroom. 9. How much did the blackboard in your school- room cost at 18^ a square foot? 255. In your drawings, let 1 in. represent 2 ft. Draw : 1. A square that will contain 16 sq. ft. 2. A rectangle that will contain 16 sq. ft. 3. A square that will contain 36 sq. ft. 4. A rectangle that will contain 36 sq. ft. 5. Find the perimeter of each of your figures. BILLS AND ACCOUNTS 177 BILLS AND ACCOUNTS 256. 1. Study the bill given in Sec. 78, p. 82. 2. A bill must always show the date of the trans- action. What is the date of the transaction re- ferred to in the bill in Sec. 78 ? 3. The debtor is the party who buys the goods. Who is the debtor in the bill referred to above ? 4. The creditor is the party who .sells the goods. Who is the creditor in the bill referred to above ? 5. An item is a separate debit or credit made in a bill. How many items are there in the bill referred to above ? 6. How many items are there in the bill in Sec. 257? 7. Name the debtor and the creditor in the bill in Sec. 257. 257. Henry Love, A RECEIPTED BILL Oakland, Cal., June 30, 1904. Bought o/ James Roland. 5 lb. sugar 2 caus tomatoes 2 lb. coffee .05 .10 .40 Received Payment, James Roland. .25 .20 .80 2^ Make similar bills, using regular bill paper. 1st Bk Aritii — 12 178 MULTIPLICATION AND DIVISION LONG DIVISION 258. When all the steps in division are written, the process is called long division. Divide 173 by 3. Model: 57f 8 is contained in 17 five times. 3)173 Write 5 in the quotient above the 15 7, as in short division. Multiply 3 23 by 5 and write the product under 21 17, and subtract. The remainder is 2 2. Bring down the 3 of the divi- dend and find how many times 3 is contained in 23. This is 7 times. Multiply 3 by 7 and write the product under 23, and subtract. Treat the re- mainder as in short division. CASE ONE 259. When the second figure* of the divisor is the same as or less than the first figure, as in 44, 63, 978, 658, etc. 1. Divide 2292 by 43. First, find how many places at the left of 2292 it will take to contain 43 at least one time. It will take three places. The first figure of the quotient will be in tens' ,, ^O N^nr^r> ^i^P 1- 4 is contained in 22 Model: 43)2292 n f. • , .. ey^r five times with 2 remainder. ~TT^ This remainder, with the next -190 figure of the dividend, is the ^r^ dividend of 3, the second figure of the divisor. The dividend of * In 63 regard 6 as the first fin:ure and 3 as the second figure. LONG DIVISION 179 3 is 29. Is 3 contained in 29 as many as 5 times ? If it is, 5 is the trial quotient figure. 3 is contained in 29 as many as 5 times. Write 5 in the quotient above 9. Step 2. Multiply 43 by 5, and write the product under 229. StepZ. Subtract 215 from 229. The remainder is 14. As this remainder is less than the divisor, the trial quotient figure is the true quotient figure. Step 4. Bring down the next figure of the dividend. The new dividend is 142. Repeat Step 1. 4 is contained in 14 three times with 2 remainder. As 3 is contained in 22 as many as 3 times, 3 is the trial quotient figure. Write 3 in the quotient. Repeat Step 2. Multiply 43 by 3 and write the product under 142. Repeat Step 3. Subtract 129 from 142. The remainder is 13. Treat the remainder as in short division. 2. Divide 1806 by 43 ;'V877 by 96 ;'\877 by 82. 3.^' Divide 2806 by 65 ;' 5927 by 97 ; 5927 by 51. 4.^^'Divide 16,108 by 72; 48,191 by 85; "45,960 by 71. 5. Divide 2115 by 43. First, find how many places at the left of 2115 it will take to contain 43 at least one time. Model : 49^3 Step 1. 4 is contained in 21 43)2115 five times with 1 remainder. As 172 3 is not contained in 11 as many 395 as 5 times, write 4 as the trial 387 quotient figure, and continue as 8 in the preceding exercise. - 180 MULTIPLICATION AND DIVISION 6. Divide 6748 by 76 ; 8482 by 54 ; 7667 by 99. 7. Divide 4084 by 72; 2094 by 93 ; 6456 by 77. a Divide 1243 by 22; 4527 by 33; 5468 by 76. 9. Divide 9247 by 95. q 9 is contained in 92 ten times. Model: 95)¥247 Use 9 as the trial quotient figure. Complete the division. 10. Divide 7492 by 77 ; 6230 by 65 ; 9348 by 98. 11. Divide 6258 by 66 ; 5213 by 55 ; 8570 by 87. 12. Divide 5234 by 54 ; 7485 by 76 ; 8775 by 88. 260. When the second figure of the divisor is the same as or less than the first figure^ the trial quotient figure may he found as follows : 1. If the second figure of the divisor is contained in its dividend * as many times as the first figure is contained in its dividend^ use this quotient figure as the trial quotient figure, 2. If the second figure of the divisor is not contained in its dividend as many times as the first figure is contained in its dividend^ use as a trial quotient figure one less than the quotient figure obtained by dividing by the first figure of the divisor. This unll he found to he the true quotient figure. 3. // the first figure of the divisor is contained in its dividend 10 times^ use 9 as the trial quotient figure.^ * The number formed by annexing the next figure of the dividend to tlie remainder left after dividing by the first figure of the divisor is the dividend of the second figure of the divisor. t The pupils should become perfectly familiar with these facts through illustration. LONG DIVISION 181 With divisors of two places, the trial quotient figure obtained as indicated in 1 and 3 will be found to be the true quotient figure. Often with divisors of more than two places, the trial quotient figure will be found to be one more than the true quotient figure. 261. Written Exercises. Examine the divisors used in these exercises. Divide : 1. 51,913 by 54. 12. 86,734 by 85. 2. 65,760 by 87. 13. 67,863 by 75. 3. 51,596 by 92. i4. 54,326 by 64. 4. 61,640 by 76. 15. 92,147 by 53. 5. 51,594 by 96. le. 92,147 by 534. 6..*83,756by 43. 17. 54,326 by 647. 7.J. 98,765 by 21. la 94,245 by 736. 8.i 45,637 by 86. 19. 67,863 by 754. 9. ^^94,245 by 73. 20. 67,321 by 959. 10.5.87,653 by 54. 21. 51,504 by 967. 11. 67,321 by 9^. 22. 45,637 by 865. 262. Written Exercises. Examine each divisor before using it. Divide each of the following by 87, 94, 63, 72, 55, 22,31,44,33,652^773,940: 1. 93,456. 3. 54,943. 5. 19,831. 7. 40,572. 2. 67,342. 4. 86,425. 6. 24,753. s. 98,345. 182 MULTIPLICATION AND DIVISION 363. Written Problems. 1. There are 52 weeks in one year. How many years are there in 468 weeks ? 2. A man bought a carload of cattle at $32 each. He paid $800 for the carload. Find the number of cattle in the car. 3. A train travels at an average speed of 42 mi. an hour. How many hours will it take it to travel 1000 mi. ? 4. A hardware merchant paid $21 each for stoves. The amount of his bill was $315. How many stoves did he buy ? 5v A dealer bought a carload of horses at $95 each. He paid $2565 for them. How many horses did he buy ? ___..._._.«.«-«««-.i.--------— -—- 6. A merchant paid $3570 for some carriages at $210 each. How many carriages did he buy? 7. At $ 65 a month J in how many months will a clerk earn $ 975 ? 8. At 55^ a yard, how many yards of cloth can be bought for $ 8.25 ? (Change to cents.) 9. At 75^ each, how many tickets must be sold to amount to $15 ? 10. There were 42 children at a school picnic. The expenses of the picnic were $ 10.50. Find each one's share of the expenses. LONG DIVISION 183 - CASE TWO 264. When the second figure of the divisor is greater than the first figure, as in 37, 28, 287, 596, etc. Divide 1734 by 47. o When the divisor is 47, 48, or 49, M . dTvPT^J ^^® ^ ^^ ^^® divisor to find the trial quotient figure. 5 is contained in 17 three times. Use 3 as the trial quotient figure. Com- plete the division. When the second figure of the divisor is 7, 8, or 9, the number to be used as a divisor to find the trial quotient figure is determined as follows : When the divisor is 47, 48, 49, 475, etc.^ use 5 as a divisor. When the divisor is 57, 58, 59, 584, etc.^ use 6 as a divisor. The trial quotient figure thus obtained is sometimes one more or one less than the true quotient figure. 265. Written Exercises. Before dividing, state what divisor will be used to find the trial quotient figure in each of the following. Divide : 1. 54,321 by 19. o^ i. 71,067 by 38. 13. 43,907 by 275. 2. 54,796 by 38. 5 8. 30,402 by 49. i4. 69,087 by 594. 3. 12,345 by 69. t 9. 42,796 by 59. 15. 87,906 by 178. 4. 43,697 by 28.> 10. 74,908 by 67. 16. 38,690 by 576. 5. 43,345 by 18. 11. 43,768 by 57. 17. 43,234 by 374. 6.^54,678 by 57. 12. 67,098 by 19. is. 41,908 by 490. 184 MULTIPLICATION AND DIVISION 266. Divide 1085 by 25. ^ 2 is contained in 10 five times. MonFT • 2^ yTTTF^ ^^^ ^^® ^^^^ than this quotient as a trial quotient figure. 4 is the trial quotient figure. Complete the division. When 23-26, 34-36, 45, 46, and 56 are used as divisors^ or as the first two figures of divisors, use as a trial quotient figure one less than the quotient obtained hg dividing the first figure^ or figures^ of the dividend by the first figure of the divisor. The trial quotient figure thus obtained is sometimes one more or one less than the true quotient figure. 267. Written Exercises. Divide each by 25, '35, 46, 56, 243, 462, and 358: 1. 9875. 7. 9356. 13. 6532. 19. 9530. 2. 2675. 8. 3640. i4. 3256. 20. 4780. 3. 3650. 9. 1884. 15. 7890. 21. 8020. 4. 4563. 10. 3657. is. 5231. 22. 6510. 5. 1895. 11. 7627. 17. 6423. 23. 1560. 6. 2750. 12. 1840. 18. 5768. 24. 7090. 268. Written Exercises. Divide each by 65, 49, 36, 245, and 684 : 1. 547,659. 5. 634,237. 9. 120,500. 2. 134,652. 6. 845,178. 10. 887,945. 3. 347,865. 7. 342,156. 11. 674,109. 4. 937,311. a. 240,100. 12. 832,674. LONG DIVISION 185 269. When such numbers as 13, 14, 15, 16, 134, 149, 157, etc., are used as divisors, the trial quotient figure can not be determined by any definite rule ; but by the following method a trial quotient figure may be found that will seldom vary more than one from the true quotient figure. Divide 1067 by 14. _ Use 2 as a divisor. 2 is con- ^ .c^TTTTTs^ tained in 10 five times. Add 2 to Model: 14)10b7 ^, . ^- ^. ^ . i x- . x^ ^ this quotient lor a trial quotient ng- ure. 7 is the trial quotient figure. Complete the division. When 13, 14, 15, 16, 138, etc.^ are used as divisors, divide the first figure, or the first two figures, of the dividend hy 2, a7id add 2 to the quotient thus obtained for a trial quotient figure, unless the quotient figure can he determined readily hy inspection. 270. Written Exercises. Divide : 1. 95,478 by 15. 6. 13,468 by 16. ii. 84,675 by 138. 2. 87,345 by 14. 7. 23,410 by 15. 12. 12,432 by 146. 3. 11,745 by 16. 8. 46,098 by 13. 13. 90,543 by 164. 4. 10,000 by 13. 9. 10,710 by 14. i4. 14,500 by 159. 5. 20,348 by 14. 10. 12,000 by 15. 15. 15,234 by 163. 271. Written Exercises. Divide each by 95, 79, 36, 16, 425, 386, and 145 : 1. 548,674. 3. 240,575. 5. 450,100. 2. 427,658. 4. 318,925. 6. 987,689. 186 MULTIPLICATION AND DIVISION 272. Written Exercises. Divide : 1. 25,678 by 38. 6. 23,670 by 15. hi. 37,896 by 376. 2. 17,408 by 19. 7. 10,682 by 14. '12. 51,678 by 645. 3. 62,389 by 83. a 25,678 by 38. 13. -10,682 by 210. 4. 37,896 by 52. 9. 17,408 by 27. i4. 12,367 by 144. 5. 51,678 by 57. 10. 62,389 by 88. j 15. 98,765 by 990. 16. There are 5280 ft. in oiie mile. Reduce 5,786,968 ft. to miles. 17. There are 365 da. in one year. Reduce ^63,475 da. to years. 18. At 15^ a gallon, how many gallons of oil can be bought for $4.50? 19. A bushel of wheat weighs 60 lb. A cental is 100 lb. How many bushels are there in 18 centals of wheat ? 20. A barrel of flour contains 196 lb. How many barrels can be filled from 6272 lb. of flour ? 21. At $ 75 an acre, how many acres of land can be bought for $ 3000 ? 22. If a man saves $ 45 a month, in how many montlis will he save enough to buy a farm worth $3150? 23. There are 144 sq. in. in one square foot. Re- duce 11,520 sq. in. to square feet. 24. There are 320 rd. in one mile. Reduce 1920 rd. to miles. WRITTEN PROBLEMS 187 273. Written Problems. 1. A farmer planted an orchard of 240 trees. There were 24 trees in each row. How many rows were there in the orchard ? 2. Find the cost of 26 cans of oil at $1.10 a can. 3. A trader paid 22 ^ a dozen for eggs. He bought $8.80 worth. How many dozen did he buy? 4. A car made 8 trips in a day. On an average the car carried 85 people each trip. The fare was 5)^. What were the total receipts for the day ? 5. If school is in session 5 hours a day for 200 days in the year, how many hours of school are there in a year ? 6. A boy's salary is .$20 a month. He has been paid $120. How many months has he worked? 7. A boy puts $ 8 in a bank each month he works. He has $176 in the bank. How many months has he been working ? 8. ' A farmer sold his hay at $ 9 a ton. He received for his crop $ 1 800. How many tons of hay did he sell ? 9. ^ A farmer sold his wheat at $.85 a bushel. He sold 78 bushels. How much did he receive for his wheat ? 10.^- A merchant buys flour at $5 a barrel and sells it at $6. How much does he gain on each barrel? $ 1 is what part of $ 5 ? 11. A boy bought 30 papers at the rate of 2 for 5^. He sold 25 of them at 5^ apiece. The other 5 he gave away. Did he gain or lose, and how much ? CHAPTER V FRACTIONS AND DECIMALS COMPOUND NUMBERS AND REVIEWS ADDITION OF FRACTIONS 274. Written Exercises. One boy and 1 boy and 1 boy are — boys. One half and one half and one half are — halves. -| are equal to — ^. Add : 4^ Add the fractions first : | + J + 1 are f 5 21 which are equal to 1^. Write the ^ under 2-| the colunm of fractions and add the 1 9 J to the column of whole numbers. 275. Add the following : a 6 c d e / 9 h / H 8 9 H 7 74 7 H 6i ^ 5 8 H H 6 51 6 51 H H 2* % H 9 6| 2 8+ 9 ii H 7i 8 81 9 n n 276.* 1. Find by addition the cost of 5 yards of cloth at 41^ a yard. 2. How far is it around an oblong that is 8^ in. long and 5{> in. wide ? Draw the oblong. * Dictate additional problems of a similar character. 188 StBTRACTlOK OF FRACTIONS 189 277. Written Exercises. Review Exercise 90, p. 88. a 6 c d e f 9 Ji H H H H 4f 64 ^ 7f 54 5* 5-1 6 n 54 H 94 64 n 6 7 5| 5 H 84 H H 7f 84 ^ 6 H 74 ^ 81 8 94 4f ^ 5f 84 2 H 9 6 H 8 6 74 1. Find by addition ^tlie distance around a room that is 4| yd. long and 3^ yd. wide. 2. Find by addition tlie cost of 6 yd. of cloth at 5|^ a yard. SUBTRACTION OF FRACTIONS 278. Written Exercises. * 1. Subtract 21 from 41. 2. Subtract 2 from 5J. Model : Model : 41 1 and make ^. -24 2 2 and 2 make 4. 5^ and ^ make -|. -2 31 2 and 3 make 5. Solve: 3a 6 c ci 41 5i 61 4| -21 -2 -31 -21 e ' f g h i 4| 4| 71 5| 7| -31 -2| -21 -3i -31 4. a 6 c 14| 231 24| _6i -18 -181 - d e f g 26| 301 401 45| . 9 -161 -10 -28J Find the sum of each of the above exercises. 190 FRACTIONS AND DECIMALS 279. Written Exercises. 1. From 27 subtract 13i. Model : 27 Add | to the minuend. J and i -13^ are |. Add 1 to o. 4 and o are 7. 13 J 1 and 1 are 2. Subtract : a h G d e f g k i 2. 4 5 8 6 3 10 25 15 24 2i 1| 3| 4| 2| _4J 12| Jl 20^ a b c d e 3. 5 7 10 8 8 a 6 c d e 4. 35 37 67 34 30 28| _9| 18| 10| lOi 5. A woman bought 8 yd. of cloth. She used 2^ yd. in making a waist. How much of the cloth had she left ? 6. A grocer bought 9 doz. eggs. He sold 5^ dozen. How many dozen did he have left ? 7. A girl bought 10 yards of lace. She used 3| yards to trim a dress. How many yards had she left? 8. A girl bought 2 pieces of ribbon. One of the pieces was 6^ yards long and the other was 4^ yards long. How many yards were there in the two ribbons ? / 9 h 9 25 34 4| 12f 151 / 9 h 40 30 40 20| lor 19| ADDITION AND SUBTRACTION OF FRACTIONS 191 280. Written Exercises. 1. From 9^ subtract 4f . Model • 9^ Since the fraction in the minuend _42 ^3 4 Addl is less than the fraction in the siib- 2 trahend, add f or 1 to the fraction ^ of the minuend, f and -| L to 4. 5 and 4 are 9.^ are i. a h c d e f 91- 71 81 141 22i 341 g 401 -^ -2| -H -4f -15| -17f -lOf 9| 7| ■ 8| 12* 27i 341 17i -4* -^ -4f -3i -18 -26| -9f 4. Find the sum of each of the above exercises. 5. From a piece of cloth containing 7^ yd. of silk a merchant sold 3|- yd. How many yards re- mained ? 6. A grocer bought 36 doz. eggs. He sold 4^ doz. to one customer and 3f doz. to another. How many dozen did he sell to both ? 7. A girl bought 6^ yd. of lace. She used 2| yd. to trim a dress. How much lace had she left ? * After the pupils have become familiar with this method, they may be taught to subtract the fraction of the subtrahend from 1, and to add the difference to the fraction in the minuend, thus: f and \ make 1. -J- (the difference) and \ (the fraction in the minuend) make f , the fractional part of the answer. 192 FRACTIONS AND DECIMALS 281. Written Exercises. a b C d e / 9 h 1. 51 6| 6i 5f H 5i 6i H 7| n 5i 2f n 8^ 5i 5 H 8| 2 8| 8| 9i 8 9f H 5f 4 9| 7i 7 9| 81 2. 5f 6| 6* 81 5| 6f 5| n H 3^ ■n 5| 6i ^ 8f 6f 4f 5| H 6| 8 H 2| 8 H 8i H 8| 9 8 3 9 Subtract ,: • 3. 61 6i ^ ^ H 4f 7f 6| 4f 2| If If H 1* 2f 1* 4. ^ 6f 8* 9f 6f n 8f 14i ^ 4 H 4i 2* H 3f 9f 5. A girl's weight on June 15th was 94| lb., and on August 15th was 102^ lb. How much did she gain in two months? 6. Alice bought 12 yd. of lace and used 8f yd. to trim a dress. How much of the lace had she left ? 7. Find the sum of 2f lb., 6f lb., and 8^ lb. 8. Find the sum of 3^ yd., 6| yd., and 8f yd. DEFINITIONS 198 282. Oral Exercises. 1. In the fractions f, f , ^, and ^, the unit of meas- ure is ^. The 5 shows into how man}^ equal parts the quantity is divided. It is called the denominator of the fractions. It names the equal parts. The 2, 3, 1, and 4 tell the number of equal parts taken, or the number of times the unit of measure is taken. The upper term is called the numerator of the fraction. 2. In 1^, 6 is the denominator. It shows that the quantity is divided into 6 equal parts, or into sixths, and that the unit of measure is ^. The numerator is 4. It tells the number of equal parts taken, or the number of tunes the unit of measure, J, is taken. 3. A fraction whose numerator is less than the denominator is called a proper fraction. 4. A fraction whose numerator is equal to or greater than the denominator is called an improper fraction. 5. Name the numerator, the denominator, and the unit of measure in the following. Tell which are proper fractions : f, f , |, f , |, f , f , f, |. 6. Such numbers as 8, 7, 4, 25, etc., are called integers. When a number is composed of an integer and a fraction, it is called a mixed number. 8| is a mixed number. It is expressed in two units of meas- ure. The 8 is expressed in ones, the | in fourths. It may all be changed to fourths. There are f in 1. In 8 there are ^-^-. ^ and J are ^-^. 1st Bk Aritii — 13 194 FRACTIONS AND DECIMALS REDUCTION OF FRACTIONS 283. To change a mixed number to an improper fraction, multiply the whole numher by the denomina- tor of the fraction, add the numerator, and ivrite the sum over the denorninator of the fraction. 1. Change 6| to an improper fraction. Model : 5 x 6 = 30 ^^. ^^^^^^ ^^^ ^ *^ ^^^^'' 30 , 4__34 multiply it by 5. 5 times 6 "^^^""^ is 30. 2. Change the following mixed numbers to im- proper fractions: ^, 9|, 7^, ^, \^, 15|, ^. 3. Write ten mixed numbers and change them to improper fractions. 4. Change to improper fractions : Tf, 6|, 3f , 4|-. 284. To change an improper fraction to a mixed number, divide the numerator by the denominator. 1. Change ^ to a mixed number. 61 Model : .vo^ Divide 25 by 4. 2. Change the following improper fractions to mixed numbers : ^, ^, ^^ J^, J^, J^. 3. Change to mixed numbers: ^, ^, l|^, ^^-. 4. Write ten improper fractions and change them to mixed numbers. 5. Find the sum of the following by adding the numerators together. Reduce the answer to a mixed number: |, ^, ^, J^^ ^, l REDUCTION OF FRACTIONS 195 285. 1. Show in a similar way :i = |, l = |,i = ^-^, 2. Show in a similar way : i = f ? i = i%? i = o"- 3. Change to 12ths : ^, \, ^, ^. 4. Which is the larger, and how much, \ ov ^-^1 5. The fractions ^, f, f , and ^^ are alike in value. They differ in /orm. 6. Can you add the following fractions as they stand: J ft., | ft., and f ft.? 7. Can you add the following fractions as they stand: 1 ft., l ft., and l ft.? 8. Can you add the fractions in Question 7 if they are changed to inches? 9. Can you add the following fractions : ;^ ft., ^ doz., 1 gal.? Is there a common unit to which they can be changed ? 10. Multiply the numerator and the denominator of ^ by 4. The answer is — . Has the value of the fraction been changed ? 11. Multiply the numerator and the denominator of 1 by 4. The answer is — . Has the value of the fraction been changed ? 196 FRACTIONS AND DKCIMALS 286. Oral Exercises. 1. Change f to 12ths. Model: 3 is contained in 12 four 1 = i| ; 1=1%; f = 3-%. times. 4 times 2 is 8. 2. When I is changed to 12ths, the denominator is made 4 times as large. We know this because 3 is contained in 12 four times. The numerator must also be made 4 times as large, so it is multiplied by 4. 3. Change to 12ths : 4. Change to 18ths : 5. Change to 24ths : 6. Change to 20ths : 7. Change to 36ths : a Change to 30ths : 287. Written Exercises. Change to 12ths and add Model A : Model B : h h h h h h h h h h h h h f. h h 6' h 1% h A- 4 5"' 3 h iV h h A- h 3 4' h T2? h h 9- h h A' A> h i' 2. 3. 4. i=A * 2 i i l=A 1 8 i i i=A h 6 i i f=A 1 9 i i B 2 t i Add the fractions in Exercises 3-8, Sec. 286. * REDUCTION OF FRACTIONS 197 288. Written Exercises. 1. Find the sum of Sf, 6|, and Gf. Change the fractions to 12ths and add the sum of the fractions to the sum of the whole numbers. Model A: Model B: 8f = 8M 8| 10 6| = 6A H 8 6f = 6A 2A = 221 6| 9 2 221 H= = 2A = 2i Change to 12ths and add : 2. 3. 4. 5. 6. 7. a 8f 5| 7i 5* ^ 8A If 6f 61 9f 8f n 7f 9| 7| 71 2i 7f 8A 6i 7J H ^ 5f 9A 4f 4f 5f 54 3| 6f 8* 5* 3f 6i i)| 81 7J 7i H 2| 8f 289. 1. The fraction f may be changed to ^ by dividing the numerator and denominator by 4. This does not change the value of the fraction. 2. Remember : If the numerator and the denomi- nator of a fraction are divided by the same number, the value of the fraction is not changed. 3. The fraction |^ is not in its lowest terTus because both the numerator and the denominator may be divided by a number that will change them to smaller numbers without changing the value of the fraction. What is the number ? 198 FRACTIONS AND DECIMALS MEASURES 290. 1. The exact measures of 12 ft. are : 1 ft., 2 ft., 3 ft., 4 ft., 6 ft., and 12 ft. 2. The exact measures of 18 ft. are : 1 ft., 2 ft., 3 ft., 6 ft., 9 ft., and 18 ft. 3. 2 ft., 3 ft., and 6 ft. are each exact measures of 12 ft. and 18 ft. They are common measures of 12 ft. and 18 ft. 4. 6 ft. is the greatest measure that is common to 12 ft. and 18 ft. It is the greatest common measure of 12 ft. and 18 ft. 291. Find the exact measures of: 1. 15 gal. 4. 16 qt. 7. $26. lo. 18 in. 2. 20 gal. 5. 24 pt. a $30. ii. 28 da. 3. 36 ft. 6. 40 gal. 9. $48. 12. 10 ft. Find the common measures of : 13. 12 and 18. 16. 16 and 36. 19. 24 and 36. 14. 24 and 30. 17. 10 and 40. 20 30 and 36. 15. 36 and 48. la 12 and 48. 21. 14 and 28. 292. Keduce to lowest terms ; 1 6_ \2.0 JL 14 3 2. 16. A 2J. 42'3_^ 30 40 6 4 •*•• 36? 36? 36? 36' 36? 36* *' 36' 6lF? 42? 4ir? 64? T'J* 32 14 1 « 3^6? 49? "?rr- 48 16 24 ¥9? T2? ¥4? 4^- 7- What is meant by the greatest common measure of two or more quantities ? Til is is generally known as the greatest common divisor, or greatest common factor of the quantities. o 6 4_' 18. 2 0. 12.1_6 q 8 6 3 54 32 14 18 -=• 24' 24? 24? 24' 24' 24* ^' 24' "ST? 63? 3^6? 4 9? IT « 12 3 2 40 ^'3 6 16 44 ^ 8 2 1 2 8 4 8 16 2 3- 4ir? 4¥? ¥¥? ¥¥' 4¥' 4¥- ^ 48' SS"? MEASURES 199 293. Oral Exercises. — Change ! to improper fractions : 1. 3f 6. 9f 11. ^ 16. 6f 21. 8J, 2. 5| 7. 8| 12. 9| 17. 5| 22. 7^2- 3. 6i 8. 5J 13. 8i 18. 2| 23. 63^ 4. 7f 9. 6f 14. 4f 19. 5| 24. SjJj 5. 8f 10. 7| 15. 3f 20. 7i 25. 8A 294. Oral Exercises. Change ! to whole or mixed numbers : 1. ¥ ft. 6. \« lb. 11. J^ft. 16. fft. 21. %^^ 2. yin. 7. ^qt 12. .2;tda. 17. |yd. 22, $V- 3. ^da. 8. Jf ft. 13. ^ mi. 18. V in. 23. $J# 4. ¥yd. 9. -2^ yd. 14. ¥pt. 19- J^mi . 24. 1-3^ 5. -V- da. 10. y rd. 15. J^ lb. 20. ¥ft- 25. $¥ 295. Oral Exercises. 1. ift. = A ft. 5. 1 ' ia. = f^ da. 9. f ft. = f^ft. 2. |y«i.= i'2 yd. 6. 1 ( ia. = 2°4 da. 10. t ft. = ^ft. 3. f mi.= f^mi. '• f ' 3a. = ^ da. "•A ft. = A ft. 4. ift. = ^ft. 8. 1 ( ia. = ^ da. 12. 1 ft. = /oft. 296. Oral Exercises. Reduce to lowest terms : 1. ^ 2. 3. M 5. 24 60 9. 1 120 13. 90 900 17. M 21. \\ li 6. fl 10. M 14. e 18. W 22. tVi f^ * 7. T2¥ 11. f* 15. M 19. M 23. 10 8 M 8. ro7 12. ft 16. M 20. 4 4 55 24. ^ 200 FRACTIONS AND DECIMALS FACTORS AND MULTIPLES 297. Oral Exercises. 1. Numbers that are exactly divisible by 2 are even numbers. 2. Name the even numbers to 30. Numbers that are exactly divisible by 2 end in what figures ? 3. Numbers that * are not exactly divisible by 2 are odd numbers. 4. Name the odd numbers to 30. 5. Some numbers cannot be divided by any whole numbers except themselves and 1 without leaving a remainder. These are called prime numbers. The following are prime numbers : 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. These can be divided only by them- selves and 1 without leaving a remainder. 6. A factor of a number is one of two or more numbers which multiplied together will make the number. 6 and 5 are each factors of 30. 2, 3, 4, and 6 are each factors of 12. 7. A common factor is a common unit of measure. 6 is a common factor of 12 and of 36, because both of these numbers are exactly divisible by 6. Name the common factors of 24 and 36. To reduce a fraction to its lowest terms, divide hot] I numerator and denoyninator by their common factors. FACTORS AND MULTIPLES 201 298. Oral Exercises. 1. The multiples of 2 are : 2, 4, 6, 8, 10, 12, etc. 2. The multiples of 3 are : 3, 6, 9, 12, 15, etc. 3. Which of the above numbers are multiples of both 2 and 3 ? 4. Numbers that are multiples of two or more numbers are called common multiples of the numbers. 6 and 12 are common multiples of 2 and 3. 5. The least common multiple of 2 and 3 is 6. 6. Find the least common multiple of 3 and 5. 7. Find the least common multiple of 6 and 8. 8. Find the least common multiple of 6 and 9. 9. In the fractions f and f , the least common mul- tiple of the denominators is 12. It is called the least common denominator of the fractions. 299. Written Exercises. 1. Change to least common denominators and add : abcdefgh.ijk I 2 4 5 4 1 3 2 2 7 1 4 5 3 5 6 7 2 8 9 5 8 9 5 T2 3 3. 2. 3. 3. 5 1 5 3. 3. JL 3. £ ^ 3^ ± A _6^ ^ 6_ ± ± _8^ 4 2. What' i^tne' least common multiple of 2, 3, and 4? 3. What is the least common multiple of 3, 4, and 5? 4. What is the least common multiple of 6, 4, and 8? 202 FRACTIONS AND DECIMALS 5. Name the prime numbers below 10. 6. Find the least common multiple of 7 and 9. Since 7 is a prime number, the least common multiple of 7 and 9 is their product. 7. What is the least common multiple of 7 and 8 ? Of 7 and 6 ? Of 5 and 7 ? Of 5 and 9 ? 300. Written Exercises. 1. From 5f subtract 2|-. Model A : 5f = 5^^ Model B : 5f 1 8 -2| = 2^ - 2||9 Add ^ to the fraction of the minuend, making it f f . ^ and ^ are ff . Add 1 to 2. 3 and 2 are 5. Solve : abcdefgh 2. 6f 5| 6i 7| 8i 6f 81 9| -41 -2| -41 -5i -5| -4| -4§ -6^ 3.15^ 19f ISf 9f 121 7^^ 6|- 5h _8i -9| -7f -4^ -8^ -21 -4A -4f 4. 18f 16f 28 43f 26f 30 30| SOf -S^. -4i -7| -7_ -4f -^1 -9^ -9f Add each of the above exercises. ORAL EXERCISES 203 301. Oral Exercises. 1. Change to improper fractions : 3f , 8|, 9f , 7f . 2. Change to whole or mixed numbers : -^^ -^, 23 60. 7 ? '9 • 3. Reduce to lowest terms : ff , If? it? if? f p 4. What are the common factors of 12 and 8 ? 5. What is the least common multiple of 12 and 8 ? 6. Change | to 30ths ; | to 40ths ; f to 24ths. n !_»'_. !—_£_. 3. .__»!_. 1 — _£L_ • 2 __ a- . '• 2~T0 0? 4"~10 0' 4~1005 5""100? 5"~100> 3 3! .4 g ;S""T0(5 ' 5 "" TWO' 8. Findfof $12; fof $12; fof $12; |of24hr. 9. 6 is fof—; 8 isfof — ; 9 is fof—; 12 is fof—. 10. Name the prime numbers between 1 and 30. 302. Written Problems. 1. A girl bought 8| yd. of ribbon. She used 5f yd. How many yards had she left ? 2. James weighs 84f lb. and George weighs 97 lb. How much heavier is George than James ? 3. What is the sum of 6| yd. and 8f yd. ? 4. A man owned 247f acres of land. He sold 122|^ acres. How many acres had he left ? - 5. The sides of a field are 271| rd., 290 rd., 175 f rd., and 180 J rd. Find the distance around the field. 6. Find the distance around a room that is 14J ft. long and 9| ft. wide. 204 FRACTIONS AND DECIMALS 303. Written Exercises. Add: a 6 c d 137^ 66f 997J 54^ 142tV 88| 885| 68,^0 183/^ 99j^ 6673-V 88,V 988| 88* 832| 99A ST'JtV 991 238^3^ 65M 777H 78A 9651 76f 304. 1. Harry weighs 72^ lb. George weighs 2| lb. more than Harry. Find the weight of George. 2. Find by addition the cost of 8 yards of cloth at 12|^ a yard. 3. A tailor had a piece of cloth containing 24| yd. From this piece he used 3f yd. to make a pair of trousers. How much of the piece remained ? 4. A grocer bought sugar -at 4f ^ a pound. He sold it at the rate of 18 lb. for $1. How much did lie receive per pound for it ? Find his profit per pound. 5. At 14^^ a pound, what is the cost of a turkey that weighs 8 pounds ? 6. At 12^^ a pound, what is the cost of a roast that weighs 8 pounds ? 7. After selling 3^ yd. of ribbon, there was left 9| yd. Find the length of the piece before the sale. 8. What number subtracted from 12 leaves 9 ? What number subtracted from |- leaves J ? 'J^ WRITTEN EXERCISES 205 305. Written Exercises. 1. Reduce to a common denominator : ^{ f , |^, j^q . 10 is a multiple of 5 ; 4 is a multiple of 2. There- fore, we need to find only the common denominator of 4 and 10. This will contain all of the denomi- nators. Why ? 2. Reduce to a common denominator and add. First determine which denominators need not be considered. / 1 i f \\ 5 9 \ «j ^1 A i \ f f 4 i i * i f t ! 1 1 1 i if f 1 1 3 3 * 8 10 3 4 1 > 4 1 f f i 2 10 5 7 1 5 1 2 1 3 1 5 5 3 1 6 9 2 6 2 3 3 4 3 6 12 7 2 \% 1%. 306. M: V ^ 1 a 6 c a e / 9- 24f 67i 34| 761 691 58f 791 661 53|- 65* 84f 73f 741 83* 731 49| 47A 59f 8Vo 561 95| 94f 78tV Subtra( 68f 72| 961 871 82tV 307. 3t: a. " Jl; £." ^, d^ X JL- /» 1. 78| 96J 67f 86| 90f 65| 98f 43A -25| '80f -37f - .47|. -37f -24| -54| -17| 2. 96f 744 28f 79A 851 47| 781 98t'2 47* 36f lOf 64f 26f 16f 45f 12| 206 FRACTIONS AND DECIMALS MULTIPLICATION OF FRACTIONS 308. Oral Exercises. 1. In tlie fraction |, which figure tells the size of the parts? What does the numerator tell ? 2. How does the fractional part f compare in size with the fractional part ^ ? 3.^ What is the sum of f and f ? Of | and | ? 4. What is the 'sum of f and f and f1 f = — ^. 9. What is 3 times f ? What is 3 times | ? To multiply a fraction by a whole number^ multiply its numerator by the whole number. If the product is an iynproper fraction, reduce it to a ivhole or a mixed number. 6. Multiply : I by 5, f by 6, f by 4, f by 3. 7. Which represents the larger fractional part, iori? iorl? lori? l or i? lor^? 8. How does the length of ^ ft. compare with the length of i ft. ? i yd. with | yd. ? i ft. with j\ ft. ? 9. If the denominator of J is divided by 2, the fraction is changed to ^. |^ is 2 times J. To mul- tiply J by 2, divide its denominator by 2. To multiply a fraction by a whole number, divide its denominator by the whole number if the denominator is exactly divisible by the whole mimber. If the residt is an improper fraction, change it to a whole or a mixed number. 10. Multiply by dividing the denominator : J by 8 ; tby3; l|by9;eby6;Mby7. MULTIPLICATION OF FRACTIONS 207 11. Multiply ^1 by 9 by multiplying the numerator by 9. 12. Multiply ^1^ by 9 by dividing the denominator by 9. 13. Which of the above methods is the easier? Why? 14. When can the easier method be used ? 309. Oral Exercises. Wherever possible, divide the denominator. abed e f 1. fx5 fx3 |x3 3^x6 {^x7 ^x4 2. fx3 |x3 |x7 ^x5 ^x6 ||x8 3. |x4 |x4 |x8 ifx7 ^x9 ||x4 310. Oral Exercises. 1. What is the meaning of 4x2? 4x1? 4x1? 2. 4x| is the same as ^ of 4. ^ of 4 = — . 4x1=-. 3. 12 X i is the same as |^ of 12. 12 x | = — . 4. If 12 is multiplied by f, will the answer be greater or less than 12 ? 5. Multiply 12 by Model : l of 12 is 2 ; f of 12 are 5 times 2, or 10. 6. Multiply: 18byf; 14byf; 12byf; 16by|; 18 by I; 10 by f ; 24 by |; 16 by |; 18 by |; 16 by ff; 20 by |; 24 by f ; 30 by ^\; 25 by f ; 2 3 12 by ^ 208 FRACTIONS AND DECIMALS 311. Oral Exercises. 1. ^ of 7 may be indicated thus: -|. This is read seven thirds, or 7 divided by 3. 2. 1 of 10 = V*-, or 3f J of 9 = f , or 1 4- 3. i of 5 = J. iof9 = f. ^of4 = f. iof5 = f. 4. Find f of 7. Model : i of 7 = 21 f of 7 = 2 times 21, or 4f . 5. Solve: 9x|; 8xf; 11 xf; 8xf; 4xf. To multiply a whole number by a fraction, divide the 'whole number by the denoininator of the fraction and mtdtiply the quotient by the numerator. 312. Oral Exercises. a 6 c d e 1. 9x|- 5x| llxf 18 xi 12 x| 2. 7x| 6xf 24 xf 11 x| 27x| 3. 8xf 16 xf 12 xf 12 x^ 21xf 4. 7xf 18 x| 18 xf 30 xf 22 x,,*^ 5. 6x| 20xf 12 xf 16x| 40 X ^% 313. Oral Exercises. a 6 c d e 1. |of30 |of7 T.%of60 Hx9 20xf 2. |of40 |of5 Wof50 Ax6 15xf 3. |of35 fofS |x7 Mx6 25xt 4. 7 of 56 fof3 4x^ 21 xf 12xf 5. |of24 f of 49 fx7 18x1 11 xf MULTIPLICATION OF FRACTIONS 209 314. Written Exercises. 1. Multiply 24| by 8. Model : 24| 8 6 192 198 First, multiply | by 8. 8 times 1 = -^^- = 6 . Next, multi- ply 24 by 8. Add the products. a 2. 35fx5 b 345| X 4 c 3065x75 d 725| X 84 3. 6T|x8 725| X 3 937f X 84 423|x60 4. 95fx5 467f X 4 7841 X 23 596| X 70 5. 301x6 879f x8 986| X 46 640| X 50 315. Written Exercises. 1. Multiply 64 by 4f.. Model : 64 First, multiply 64 by 4| 38f .256' 294| Solve : f. lof64=12i fof 64 = 3 times 12^ or 38f . Next, multiply 64 by 4. Add the products. a 2. 675 x9f 6 300 X 4f c 464 X 6f d 723 X 6| 3. 864 x4f 950 X 2f 405 X 91 800 X 5| 4. 576 X 5| 375 X 8f 672 x3| 967 X 6| 5. 674 x8|- 675 X 3f 456 X 51 734 X 4f IST I'.K AKMII — 14 210 FRACTIONS AND DECIMALS 316. Written Exercises. Multiply each by 6, 8, and 9: a h c d e 1. 23 If 579| 768-1 756| 697f 2. 327f 968f 648f 748^ 764f 3. 432f 786f 975^ 654H 924f 317. Oral Exercises; Red ace to lowest terms : if 18 24 20 3-0 14 28 27 36" in 48 3.6 48 M 64 T2 d 12 144 19 TF8 56 96 14 4 50 T70 20 f 25 TFCT 40 IFT) 1^ 318. Oral Exercises. Change to wliole or mixed numbers : 1. 2. 3. 319. Written Exercises. Find the sum and the difference of each : a b C d e / !/ ¥- ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 100 ¥ ¥ ¥ ¥ fl ft 100 ¥ ¥ W 1. 97| lb. 28^ lb. 104f ft. 74i ft. 924f lb. 634^ lb. 764j3g mi. 684,-V mi. 76^ mi. 241 mi. 367f A. 145f A. 375fyd 194^ yd. 6941 lb. 375f lb. WRITTEN PROBLEMS 211 320. Written Problems. 1. Find the cost of 6 lb. of sugar at 5|^ a pound. 2. At 12^^ a pound, how much will 6 pounds of meat cost? 3. What is the cost of 3|- lb. of steak at 16^ a pound ? 4. George lives 4f mi. from the city. How far must he ride in making 2 round trips to the city ? 5. The average weight of 4 boys is 87f lb. Find their total weight. 6. Alice bought 51- yd. of lace. She used 2| yd. to trim a waist. How much of the lace had she left ? 7. If each can of milk contains of gal., how much milk is there in 6 cans ? 8. A man walked at the rate of 4 miles an hour. It took him 2| hr. to go from his home to the city. How far from the city did he live ? 9. A man had 137i A. of land. He sold 43f A. to one man and 641 A. to another. How much did he sell to both ? How many acres had he left ? 10. How much will 2| lb. of tea cost at 60^ per pound? 11. Find the cost of - 8 yd. of cloth at 121^ per yard. 12. How many rods of fence will it take to fence in a garden 14 1 rd. long and 8| rd. wide? 13. Find the cost of 8f T. of coal at | 6 per ton. 212 FRACTIONS AND DECIMALS DIVISION OF FRACTIONS 321. Oral Exercises. 1. How does the fractional part | compare in size with the fractional part f ? 2. How can the fractional part f be obtained from the fractional part ^ ? 3. How can the fractional part | be obtained from the fractional part f ? 4. If the fractional part f is divided by 2, the quotient will be y. To divide a fraction hy a ivhole number, divide its numerator hy the whole number. 5. Divide f by 2; ^ by 2;^f by 2; || by 3; f by 3. 6. Divide ff by 7; f by 4; f by 2; ff by 5; |f by 6 ; ^1 by 4. 7. Divide 4 by 4; If by 10; If by 8 ; ,-% by 3; H by 3; 1^ by 5. 8. How does the fractional part J compare with the fractional part J? 9. What part of J is | ? What part of J is f-^ ? 10. How does the fractional part ^ compare with the fractional part i ? How can the fractional part ^ be obtained from the fractional part ^ ? 11. If the fractional part ^ is divided by 2, the quotient will be — . Multiplying the denominator of a fraction by 2 has what effect upon the value of the fraction ? DIVISION OF FRACTIONS 213 322. To divide a fraction hy a ivliole number^ mid- tq^ly the denominatoi' of the fraction hy the ivhole number. 1. Divide: I by 4; I by 5; f by 7; | by 2; f by 5. 2. Divide: i by 2; f by 3; I by 3; | by 4; i by 6; iby 5. 3. Divide: I by 5; 11 by 2; f by 4; | by 3; k by 3. 4. Divide the fraction i| by 6 by dividing its numerator by 6. To divide by 6 is to find ^. Find 1 of ^^ 5. Divide the fraction ^ by 6 by multiplying its denominator by 6. Reduce to lowest terms. 6. Which is the easier method of dividing i| by 6? Why? 7. When is it easier to find the quotient by dividing the numerator by the whole number ? 8. When is it easier to find the quotient by multi- plying the denominator by the whole number? 323. Oral Exercises. Use the easier method in solving each : 1. a 6 C d e f-^3 1-4 H-5 H-6 tt-12 1-3 1^10 lf-6 H-7 M-25 1-5 f-10 lf-2 A-3 tVo-10 1^3 H-7 If -5 A-4 tVo-20 214 FRACTIONS AND DECIMALS 324. Oral Exercises. Use the easier method in sol ving each : a h C d e 1^5 f-^4 il-5 30 X fo iof72 fx3 9xf il-8 fof7 f of27 |x4 .8xf lf-5 |of2 1 of 25 |x4 Ax3 H-5 f of 9 foflO i-6 1-^x5 17x1 |of5 A of 8 f-6 M-5 lOxf |of48 A of 50 1-6 M-8 27 x| |of 12 Mof 24 325. Written Exercises. 1. Find i of 212f . Mnn-pT. ' . Jm. fi is pnnf ainpri in /^,1 f lirpp f iiTiPs witli 3 remainder. 6 is contained in 32 five times, with 2 remainder. The whole remainder is 2|. Re- duce it to ^. ^ -5- 6 = JJ. This is the fractional part of the quotient. 2. Divide each by 6 ; by 7 ; by 4 : 632f, 3451 426f, 7851, 967f, 872f 326. Oral Exercises. Divide each l)y 4 ; by 5 ; by 6 ; by 7 ; by 8; by 9 : a 6 f? rf e 1. 4| 6! 9f 7§ lOf 2. 61 8i 8f 9f 12i 3. 8| ^ 5i 8f lit 4. 7| 2f 7§ n 1% WRITTEN PROBLEMS 215 a27. Written Problems. 1. Find the cost of 2| lb. of coffee at 30^ per pound. 2. A boy had 45^. He spent f of his money for a book. Find the cost of the book. 3. A horse was bought for $120 and sold for 1| times its cost. Find the selling price of the horse. 4. There are 60 minutes in one hour. How many minutes are there in ^ of an hour ? 5. Find the area of a blackboard lOf ft. long and 3 ft. wide. 6. A blackboard containing 38J sq. ft. is 3 ft. wide. Find its length. 7. Six girls bought a box of candy weighing 1^ lb. They shared it equally. How much candy did each girl receive ? 8. Mary is in school 5^ hr. each day. How many hours is she in school each week ? 9. A woman bought 12 yd. of silk at % IJ per yard. She handed the dealer $25. How much change should she receive ? 10. A girl had 6^ yd. of ribbon. After using f yd. for a bow, how much had she left ? 11. At 12i^ a dozen, how much will 8 dozen eggs 031 ? 12. Find the area of a field 53^^ rd. lono^ and 30 rd. wide. There are 160 sq. rd. in 1 A. Find the number of acres there are in the field. 216 FRACTIONS AND DECIMALS 328. 1. Draw an oblong. Divide it into 4 equal parts. What is each part called ? 2. Show f of the oblong. Show J of |^ of the oblong. ■^ of f of the oblong is what part of the oblong ? 3. Show f of f of the oblong. | of f of the oblong is what part of the oblong ? 4. J of f is the same as f x ^, which is read J multiplied by J. It is equal to — . 5. f of f is the same as |^ x |. It is equal to — . To multiply a fraction hy a fraction, multijoly the numerators together. This product is the numerator of the answer. Multiply the denominators together. This product is the denominator of the answer. The answer shoidd be expressed in its lowest terms, 6. Multiply f by |. Model: fxi r.^% 2x4=^ 3x5=15* 7. Multiply f by f ; f by I; f by f ; f by f. 8. Multiply f by f. If we multiply as above, the answer is -^. To reduce -f^ to its lowest terms, both the numerator and the denominator must be divided by 3. The answer is f. Since there is a 3 in one of the numerators, and a 3 in one of the denominators, the reduction can take place before the multiplication. 1 no f) thus : - X ^ = ^- This is called cancellation. P 5 5 1 MULTIPLICATION OF FRACTIONS 217 329. Oral Exercises. 1. Multiply f by f ^ Cancel the common factors. Divide the 5 in ^ by 5, and the 5 in f by 5. Multiply. Model : 4 r^ 4 1 a b 2. |xf i*x| 3. ixf lfx| 4. 8^2 9^3 i-fxf X 6 2ii 21 11 13 5^12 ■8^13 T ^ 12 9 V 7 T3"^ 9 9 y 5 12 ^9 14 y 5 4 V 2 11^ 3 13 TO -^^S" X 9 '^ 8 5 V 2 6"'^ 3 3 V 1 4^8 1 x-S. 330. Written Exercises. 1. Multiply 6f by 4- 3 4- Model Solve : 62 ^3 4* 2^ ^3 24 31| iof 6f = If, or 5. 4 times 6 nets. If- f of 6f is 3 times 4 times # is #, or 2|. 3? is 24. Add these prod- 5. 3|x6i 9|x8f 4fx4| 4|x6f 5|x6f 5fx8f 6Jx4i 9fx7| 9f x 3| 4fx3f 5i X 71 lit X 3,-^ m X 4f 19fx4f lOf X 4| 218 FRACTIONS AND DECIMALS MEASUREMENTS 331. 1. Draw a line 4 ft. long. Measure it with a measure 2 ft. long. How many times did you apply the measure ? 4 ft. -h 2 ft. = — . 2. Measure a line 4 ft. long with a measure 1 ft. long. How many times did you apply the measure ? 4ft.-^l ft. = — . 3. Measure a line 4 ft. long with a measure i ft. long. How many times did you apply the measiu-e ? 4ft.^ift. = — . 4. Measure a line 8 ft. long with a measure 2 ft. long. How many times did you apply the measure ? 8 ft.^2ft. = — . 5. Measure a line 8 ft. long with a measure -J- ft. long. How many times did you apply the measure? Sft.-f-i ft. = — . 6. Divide 8 ft. by 2 ft. ; by 1 ft. ; by 1 ft. 7. . Measure 6 ft. with a 3-ft. measure ; with a 1-ft. measure ; with a |-ft. measure ; with a f-ft. measure. How many times did you apply each ? 8 How does the number of times that you applied the 1-ft. measure compare with the number of times that you applied the ^-ft. measure ? 9. If the 1-ft. measure is applied 6 times in measur- ing the length of a line, how many times must the J-ft. measure be applied to measure the same distance ? 10. How many times must the measure ^ ft. be applied to measure 6 ft. ? 6 ft. -^^ ft. = 24. ORAL EXERCISES 219 332. Oral Exercises. 1. In 8 ft. there are 16 half feet. 8 ft. ^ 1 ft. = 16. 2. In 6 ft. there are 18 third feet. 6 f t. -^- ^ ft. = 18. 3. In 6 ft. there are 18 third feet or 9 two-thirds feet. 4. How many fourths are there in 1 ? In 8 ? 5. How many thirds are there in 1 ? In 6 ? In 9 ? 6. 12ft.H-l ft. = — ; 12ft.-^ift. = — 5 12ft.^| ft. = — . 333. To divide a lohole numher hy a fraction, invert the divisor and multij)hj. 1. Divide 12 by | ■. ^ 4 d Model : Invert f tof. a b c e 2. 12^1 7-f 36 ^f 12mi.^f mi. 20-f 3. lo^f 8-t 10^1 8 ft. ^f ft. 15-f 4. 18^1 6-f 30-1 101b. -^flb. 8^1 5. 16^1 9H-f 8-f 16hr.^f hr. 7-t 334. 1. How many boxes of candy each weighing 1^ lb. can be filled from a pail containing 16 pounds ? 2. A girl bought 3 yd. of ribbon at 12^ a yard. How many hair ribbons each f yd. long can be made from it ? Find the cost of each ribbon. 3. How many rolls of butter each weighing i lb. can be made from 14 pounds ? 4. There are 5| yd. in a rod. How many yards are there in 6 rods ? 220 FRACTIONS AND DPXIMALS 335. 1. Measure 1 ft. with ^-ft. measure. How many times did you apply the measure ? 1 f t. h- 1 ft. = 4. 2. Measure |^ ft. with ^ft. measure. How many times did you apply the measure ? |^ ft. -h i ft. = 3. 3. Measure ^ ft. with ^ft. measure. How many times did you apply the measure ? |^ ft. ^ J ft. = 2. 4. Measure ^ ft. with ^ft. measure. The measure is applied ^ times. -J- ft. h- i ft. = ^. 5. If a measure ^ ft. long is used to measure \ ft., the measure would be applied ^ times. J ft. h- i ft. = ^. 6. If a measure J ft. long is used to measure ^ ft., would the measure be applied more or less than 1 time ? With your measure find what part of | ft. is used to measure J ft. ^ f t. -j- i ft. = — . 7. Measure f ft. by f ft. by changing both to 12ths : f ft. = 3^ ft. ; f ft. = 3% ft. Measure j% ft. by ^ ft. 8. Divide f by f . This may be done by changing the fractions to a common denominator and dividing the numerators, thus : f-^| = x2"^T^'^^"^^~-'^8"- 336. The following is a shorter method : 7b divide a fraction hjj a fraction, invert the divisor and midtiphj. 1. Divide | by f . Model: f-f = f x f = f, or 1|. 2. i-f 3- f-f A-l 14 _^ 7 W^8 H- -f \i-i n-f 1 r> . 8 TT--J7 1 '.\ T4" -h H-l DIVISION OF FRACTIONS 221 337. Written Exercises. a. Divide 6| by |. Model: 61 = -^^. |>^i = f x5 = |, or Sf Solve : a b C rt e 2. 6|^| 121- -f 74 . 3 '5 ^ 5 lOf-f 9f-l 3. 8|-| 18|- -i yl . 2 ^6-" 3 45f^f lU-i 4. n^i 25f -f 9f-| 7|-i 8^-1 5. 7f-| 12|- .^3 • 4 9f-A 8i-| H^h 338. Written Exercises. 1. Divide 4| by 3f . Model c ^ 42=14. 35 = 2_3_ 14^28^14 1^28 ^j. 1 _5_ Divide : a 6 C eJ 2| by 4f 8i by 2f 5J by 4f 4|. by 3| 6* by 21 9| by 31 9iby8i 2| by 3| n by 6| 5| by 7| 81 by 6f 91 by 41 5. 5|by8f 4fby3f 5| by 3^ 6| by 4i 339. Written Exercises. a b C d 1. 16f-12f 53| + 34f 8|x 4t 651^ i 2. 40f + 17f 87|-70f 9|xl0 241 X 5f 3. 56f-25l 24f + 14f 65|^ 4 401- 34| 4. 6.5f + 37f 44* - 14f 3f- 41 5f- C| 5. 18|x 6f 451 + 541 6-1-17 19f- 9f 6. 37|^ 41 691 _ 43| 56§^ 8 4|x 4f 222 FRACTIONS AND DECIMALS 340. Written Exercises. 1. Find tiie surface of a walk 12 ft. 8 in. (12| ft.) long and 6 ft. 9 in. (6| ft.) wide. 2. Find the cost of laying cement on the same walk at $ .12 J per square foot. 3. One field contains 36f A. ; another 22| A. Find the number of acres in the two fields. 4. If a train travels 560-| mi. in 12 hr., what is the average rate per hour ? ' 5. If it cost $ 36,000 to build 3^ mi. of trolly line, how much on the average did it cost to build 1 mi.? 6. What will be the cost of 16 doz. 8 eggs (16| doz.) at 16^ a dozen? 7. A man rented his farm for ^ of the crop. He received for his share of oats 360 sacks. How many sacks of oats were raised on the farm ? 341. Oral Exercises. 1. What part of 1 doz. eggs are 8 eggs ? 2. Express in dozens and fraction of a dozen : 3 doz. 5 ; 4 doz. 8 ; 6 doz. 7. 3. Express in feet and fraction of feet : 3 ft. 6 in.; 4 ft. 8 in.; 9 ft. 7 in. 4. Find the cost of 5 doz. 6 eggs at 16^^ per dozen. 5. When coal is selling at $ 8 per ton, what part of a ton can be bought for $ 2 ? For $ 4 ? For $ 6 ? 6. How many pieces of string each | ft. long can be cut from a string 6 ft. long ? WRITTEN EXERCISES 223 342. Written Exercises. 1. A farmer raised 9| tons of hay on 4 acres. What was the yield per acre ? 2. A man bought a house and a 50-ft. lot for $ 4500. If the house was valued at $ 3500, what was the value of the land per front foot ? 3. When cloth is selling at f dollar per yard, how many yards can be purchased for $ 3 ? 4. A merchant bought cloth at $ .621 a yard. He sold it at $ .75 a yard. How much did he gain on each yard ? 5. A merchant bought cloth at $ .37^^ a yard, and sold it at $ .25 a yard. How much did he lose on each yard ? 6. When hay is selling at $ 8 a ton, how many tons can be purchased for $ 12 ? $6? $9? $15? 7. Divide 7f in. by 2 ; 8f ft. by 3 ; 6| A. by 2 ; T% in- by |. 8. A woodchopper cut 12 cords of wood for $ 18. How much was that a cord ? 9. A boy has 25^ in nickels, 30^ in dimes, $3 in quarter-dollars, and $ 5 in half-dollars. How many pieces of money has he ? 10. One square inch is what part of a square foot? 11. One square foot is what part of a square yard? 12. Divide f by 2; f by 4 ; f by f ; f by 5 ; 12| by 6; 2fby5; 6| by 4. 224 FRACTIONS AND DECIMALS RATIO AND PROPORTION 343. 1. 12 apples are worth — times as much as 6 apples. 2. 4 apples are — third of 12 apples. 3. 12 apples are worth — times as much as 4 apples. 4. If 4 apples are worth 5^, 12 apples are worth — ^. 5. If 8 apples are worth 10^', 24 apples are w^orth — ^. 6. If 7 bu. of corn are worth $ 4.20, 21 bu. are worth $ — . 7. If 4 lb. of coffee are worth $1, 6 lb. of coffee are worth $ — . a If 21 yd. of cloth cost $ 2, 5 yd. of it will cost $ . 9. 121 is _ half of 25; — eighth of 100; — fourth of 50; — sixth of 75; — third of 37J-; — fifth of 621. 10. If 25 yards of cloth cost $2.12, at the same rate how much will 50 yd. of the cloth cost ? 11. 331 is of 100. 75 is of 100. 66| is of 100. 371 is of 100. 12. 66f bu. of wheat are w^orth — thirds as much as 100 bu. 13. 371 bu. of wheat are worth — eighths as much as 100 bu. RATIO AND PROPORTION 225 14. 10 brooms are worth — times as much as 4 brooms of the same. kind. 15. 1 of 6 is i- of — . 1 of 8 is i of — . 16. f of 9 apples are | of — apples. 17. t of 28 ft. are — ft. 8 ft. are | of — ft. 18. If 5 tons of hay will feed 6 horses a certain time, 15 tons will feed — horses for the same time. 19. Twelve sacks of barley will feed 6 horses as long as — sacks will feed 24 horses. 20. Working for the same wages, 5 men can earn as much in 12 weeks as 10 men can earn in — weeks. 21. The ratio of 5 men to 10 men- is the same as the ratio of 12 bu. to — bu. 22. The ratio of |^ to | is the same as the ratio of f to— . Find the cost : 23. Of 16 apples if 8 apples cost 12^. 24. Of 32 sacks of barley if 8 sacks cost $ 7. . 25. Of 25 tons of hay if 100 tons cost $ 824. 26. Of 331 yd. of cloth if 66f yd. cost$ 36. 27. Of 121 boxes of apples if 37^ boxes cost $ 27. 28. Of 10 tons of grapes if 4 tons cost $ 48. 29. Of 8 sheep if 24 sheep cost $ 45. 30. Of 11 yd. of cloth if 44 yd. cost $ 22. 31. Of 1 sq. yd. of blackboard if 1 sq. ft. cost 20^. 32. Of IJ doz. chickens if 3 doz. cost $ 12. 1st Bk Ahith — 1.5 226 FRACTIONS AND DECIMALS AREAS 344. 1. A rectangle 5 ft. long and 2^ in. wide con- tains — sq. ft. 2. A schoolroom is 30 ft. long and 26 ft. wide. The floor contains — sq. ft. 3. How many corners are there on a cube ? On a box? How many. faces are there on a cube? On a box? 4. Find the area of one face of a 3-in. cube. 5. Measure the length, width, and height of any box. Using these dimensions, find the following : the area of one of the ends ; the area of the two ends ; the area of one of the sides ; the area of the two sides ; the area of the bottom of the box ; the area of the top and bottom ; the area of the six faces. 6. Is the room you are in shaped like a box ? 7. How many faces has the room ? The floor of the room corresponds to the -^ of the box ; the ceiling of the room to the — of the box ; the sides of the room to the — of the box ; and the end of the room to the — of the box. 8. The man who plastered your schoolroom was paid a price per square yard for doing the work. If he received 21^ per square yard., how much did he receive for plastering the room ? 9. Think of a room that is 15 ft. long, 12 ft. wide, and 10 ft. high. Find : a. The area of the floor ; h. the area of the sides. DECIMAL FRACTIONS 227 DECIMAL FRACTIONS 345. 1. Divide 27 by 4. Divide 32 by 5. Read the answers. 2. Divide 27 by 10. Divide 32 by 10. Read the answers. 3. 2^ is also written 2.7. S^q is also written 3.2. 4. The period between 2 and 7 in 2.7 is called the decimal point. The fraction ^V may be written .7. Any fraction whose denominator is 10 may be so written. The form j-V ^^ ^^^ form of a common fraction. The form .7 is the form of a decimal fraction. 5. 7| is the quotient of 37 -f- 5. Is the divisor found in the quotient ? If so, where ? 6. Does the divisor appear in the quotient of 27 -^ 10? Where? 7. 3|- is the quotient. Find the divisor, the re- mainder, and the dividend. 8. The following are quotients. Name the divisors, the remainders, and read the quotients : 8|, Q^, 5^^ 6.4, 5i|o, 6.7, 50.52, 6.75, 9.38. 9. Change the following to decimals: 8^, 9^^^, 18 J^, 27,-%, 7,^, V,. 10. Read the following: 8.1, 17.5, 20.8, 9.9. 11. Write as common fractions: .6, .9, .5, .4. 12. Divide the following by 10 by placing the deci- mal point between the units' and tens' places in each: 93, 84, 935, 61, 80, 400, 405. Read the quotients. 228 FRACTIONS AND DECIMALS 346. 1. .7 is read — ; .07 is read — ; .007 is read seven thousandths. 2. Ten cents is what part of one dollar ? Write ten cents, using the dollar sign. 3. Write- one cent, using the dollar sign. 4. $.10 is how many times $.01? .10 is how many times .01 ? 5. Which is more, .8 mi., .08 mi., or .008 mi.? 6. Which is more .6 mi., .60 mi., or .600 mi. ? 7. Compare : .4, .40, .400. Compare : .4, .04, .004. ^ 10 — T0¥— 1000* To FO — TF(J — TO • 9. Express the fractions in Exercise 8 in the form of decimal fractions. 10. A decimal fraction is one whose denominator is 10 or some power of 10, as 100, 1000, 10,000, etc. 11. Read: .8, .67, .672. When the number of places to the right of the decimal point is one, the denominator is — ; when the number of places is two, the denominator is — ; when the number of places is three, the denominator is — . 12. In the decimal .7 what is the numerator? What is the denominator ? 13. What is the numerator of the fraction .05? Of .324? 14. In .72 what is the numerator ? NOTATION AND NUMERATION OF DECIMALS 229 NOTATION AND NUMERATION OF DECIMALS 347. 1. The names of the orders to the right of the decimal point are : First : Tenths' order . . ; 8 Second : Hundredths' order 67 Third: Thousandths' order .672 Fourth : Ten-thousandths' order . . . .6789 Fifth : Hundred-thousandths' order . . .67898 - Sixth : Millionths' order 678968 2. Memorize the above. Remember that four deci- mal places give ten-thousandths ; that five decimal places give hundred-thousandths, etc. 2h read a decimal, read the number ivithout refer- ence to the decimal point, and then add the name of the order of the right-hand figure of the numerator. 3. .62 is read sixty-two hundredths. It is given the name of the second order, hundredths. 4. .00062 is read sixty-two hundred-thousandths. Read .0062. 5. 6.72 is read six and seventy-two hundredths. Read : 24.52, 5.672, 6.08, 52.004, .52 oz., 6.5 oz. 6. Write in a column, with the decimal points directly below one another, and read : .272, .27, 7.62, 7.3, 4.67, 9.787, 6.72896. 7. Write in a column : four and sixty-two hun- dredths, five and sixty-five thousandths, seven and six tenths. 230 FRACTIONS AND DECIMALS ADDITION OF DECIMALS 348. 1. Find the sum of 1.27, 36.2, and 54.036. Write the numbers so that the Model : 1.27 decimal points are directly below 36.2 one another. Add as in whole 54.036 numbers. Place the decimal point 91.506 in the sum directly below the decimal point in the addends. 2. Add: 36.5,42.47, 62.367,48. 3. Add: 7.27,52.005,64.3,52. 4. Add : .05, 1.0501, 10.504, 150.41, .546. SUBTRACTION OF DECIMALS 349. 1. From 5.2 subtract 2.27. Consider 5.2 as 5.20. Subtract Model: 5.2 as in whole numbers. Place the 2.27 decimal point in the answer directly 2.93 below the decimal point in the sub- trahend. 2. From 16.7 subtract 10.25. 3. From 126 subtract 8.75. 4. How much more is 35.12 than 14.6? 5. How much less is 84.7 than 125.5 ? 6. A man owned 127.7 A. of land. He sold 27.9 A. to one neighbor and 30.5 A. to another. How many acres had he left ? FRACTIONS CHANGED TO HUNDREDTHS 231 FRACTIONS CHANGED TO HUNDREDTHS 350. 1. Memorize the following : 2^ ~"TFO "~ •^^* 4~10 0~~-^'^- 25 "" 10 ~ •^^^• JL _ 2 5 — 25 3.__75_— 75 2 — 20 — OQ t~Too — •^^- 4— loo--'^' 10 — 100 — •^^• 1 — 20—90 2 — 40 _ 40 3 _ 30 — QO 5"-roo--^^- :5^- 100 -•^'^- To-Ton^--^^- 1_10_10 3_60_A0 4_40_i0 TO-TOO--'^^- 5-T^O--'^^- TO-T¥^--^^- 1 _ 5 _ 05 4_ 80 _ CO 1 _ 2 _ (W 20 - TOO - '^^' 5 - TOO - O^- 50 - Too " •^^^• 2. Express as common fractions : .80, .75, .25, .50, .60, .40, .30, .20. 3. Express as common fractions : .05, .02, .04, .01, .10, .70, .80, .90. 4. Write with the fractional part expressed as a decimal: 6|, 71, 8f, 12^ 25 J^, 28^, ^, 3f, 14f, 221 .5. Write with the decimal part expressed as a common fraction: 4.25, 6.20, 7.75, 12.80, 15.04, 35.02, 27.75, 15.60. 6. Write as decimal fractions and add : f , \^ 3^, _3_ 3 JL 20' 5' 50* 7. Change to common fractions and reduce to lowest terms : .375, .125, .625, .875. 8. What effect upon the value of .25 has annexing a cipher to the right of it, thus : .250 ? 9. What effect upon the value of .25 has the plac- ing of a cipher before it, thus : .025 ? 10. Express in dollars and cents: $6|, $8 J, $9 J, $7-g-, $lU-j_0-, $1^4, ^IOyq-, $0^-q, $7;g^-Q. 232 FRACTIONS AND DECIMALS MULTIPLICATION OF DECIMALS 351. 1. Read the following : 5, .5, .05, and .005. 2. Compare the value of 5 and .5 ; of .5 and .05. 3. Compare : 625 ft. and 62.5 ft. ; 62.5 ft. and 6.25 ft. ; 6.25 ft. and .625 ft. 4. Moving the decimal point one place to the left has what effect upon the value of a number ? 5. Compare: .385 ft. and 3.85 ft.; 3.85 ft. and 38.5 ft. ; 38.5 ft. and 385 ft. ; 385 ft. and 3850 ft. 6. Moving the decimal point one place to the right has what effect upon the value of a quantity ? 7. What part of 22 is 2.2 ? Of 30 is 3 ? Of 3 is .3 ? 8. Multiply: 62 by 10; 36 by 10; 675 by 10. 9. Multiply: ^ by 10 ; ^ by 10 ; ^r^ by 10. 10. Multiply : .3 by 10 ; .03 by 10 ; .003 by 10. 11. Multiply in the shortest way possible : 2 by 10 ; .2 by 10 ; .4 by 10 ; 37 by 10 ; 3.7 by 10 ; .37 by 10. 12. Multiply each by 10 : .67, 5.2, .52, 6.27, 7.89. 13. Compare the value of $1.84 and $184; of $125 and $1.25. 'i4. Multiply each by 100: $6.25, $.50, $.05, $1.05. 15. Multiply 12 by ^V- Multiply 12 by .1. 16. Divide 12 by 10. 12 x .1 = 1.2. 17. Multiplying a number by -^ is the same as dividing the number by — . MULTIPLICATION OF DECIMALS 233 352. To multiply a decimal by an integer, multiply as in whole numbers. Point off in the answer as many decimal places as there are decimal places in the midtiplicand. 1. Multiply 42.35 by 8. Model : 42.35 338.80 2. 63.75x5 3. 327.42x9 4. 6.843x105 353. To multiply an integer by a decimal, multiply as in ivhole numbers. Point off in the answer as many decimal places as there are decimal places in the multiplier. 1. Multiply 367 by 8.2. Model : 367 8.2 734 2936 3009.4 2. 325x8.4 3. 863 X. 94 4. 754 x .38 354. To midtiply a decimal by a decimal, multiply as in ivhole numbers. Point off in the answer as many decimal places as there are decimal places in both mul- tiplier and midtiplicand. 1. Multiply 5.25 by 4.7. Model : 5.25 _M 3675 2100 24.675 2. 6.24x4.7 3. 95.7 X. 56 4. .875x4.5 234 FRACTIONS AND DECIMALS 355. Written Exercises. 1. 6.34x2.4 5. .937x6.3 9. .036x4.5 2. 83.6 x. 36 6. .372 X. 27 10. .004x4.05 3. 745 X. 67 7. 67.5 X. 04 11 .405 x .006 4. 8.32x45 8. 9.67 X. 003 12. 5.07 x .001 PERCENTAGE 356. 1. Read : Y^Q, .04. This may be written thus: 4%. It is then read, four per cent. Per cent means hundredths. 2. 5% is the same as -^q, or .05. 3. Express as per cent: ^f^, ^, ^, ^^^, /^o^. 4. Express as per cent: .03, .09, .12, .40, .75, .01. 5. Express as fractions: 11%, 15%, 20%, 35%, 80%. 6. Express as decimals: 4%, 18%, 7%, 75%, 10%. 7. 6% of $65 is the same as $65 x .06. 8. Find 8% of $500; of $250; of $1000. 9. Find 7% of $100; of $62.50; of $83.75. 10. To find 10% of any number, divide the number by-. 11. Find 10%of $250; of $340; of $400; of $1000. 12. 25% is the same as ^. To find 25% of a num- ber, divide the number by — . 13. Find 25% of $40; of $800; of $1200; of $4. 14. 50% is the same as — . To find 50% of a num- ber, divide the number by — . 15. Find50%of $80; of $100; of $400; of $1000. FRACTIONS AS PER CENTS 235 FRACTIONS AS PER CENTS 357. To change a fraction to per cent, multiply the fraction by 100. Reduce the product to a whole or a mixed number. 1. Change | to per cent. Model: f x 100 = ^f^ = 40. f = ^u7^. Change to per cent : J, |, |, |, ^, }, | 2. Memorize the folloioing : 1=50%. i=16t%. |=66|%. 1=33^%. i=14f%. 1=75%. 1 = 25%. i=12i%. 1=371%. 1 = 20%. ^0 = 10%. 1 = 621%. 358. Oral Exercises. Change the per cent to a fraction and find : * 1. 25% of 200 ft. 7. 50% of 800 mi: 2. 20% of $150. 8. 25% of 360 A. 3. 331% of $210. 9. 20% of 100 yd. 4. 75% of 400 ft. 10. 14f % of 70 yr. 5. 66f% of $120. 11. 121% of $720. 6. 371% of $80. 12. 10% of $950. 13. What is 20% of $375? 10% of $236? 50% of $278? 10% of 362 gal.? 25% of 640 lb.? 75% of 360 A.? 14. What is 331% of 360 A.? 66f%of $120? 14f% of $140? 16|% of $180? 371% of $800? * Supplement this exercise with oral drill until the pupils are able to find the above per cents readily by the use of their fractional equivalents. The fractional equivalents of the above per cents should be used in subse- quent exercises. 236 FRACTIONS AND DECIMALS 359. Written Exercises. 1. Merchandise is generally sold at a certain per cent profit on the cost. What per cent of profit do you think a grocer should make on tea ? * On sugar ? On strawberries? 2. If a grocer buys tea at 30^ a pound, and sells it at a profit of 33 J%, what is the selling price per pound? 3. A merchant bought a suit for $15 and sold it at a profit of 20%. What was his profit ? 4. Locate Ogden, Omaha, and San Francisco on the map. It is 844.7 mi. from Ogden to San Fran- cisco, and 1004.7 mi. from Ogden to Omaha. How far is it from San Francisco to Omaha ? How much nearer is Ogden to San Francisco than to Omaha ? 5. A merchant bought cloth at $.12 a yard, and sold it at a gain of 25%. What was his profit on each yard ? What was the selling price per yard ? 6. A clothing store advertised boys' suits worth $12 at 25% reduction. Find the amount of the reduction and the cost of a suit. 7. A dry goods store advertised a 20% reduction sale on carpets. Find the reduction per yard on car- pets that formerly sold at 60^ a yard. 8. A dealer in farm implements bought carriages at $50 and sold them at a profit of 20%. Find his profit on each carriage sold. * Discuss these and similar questions with class. INTEREST 237 INTEREST 360. 1. When one rents a house from another, how does he usually pay for its use ? 2. When one rents a farm from another, how does he usually pay for its use ? 3. When one borrows money from another, how does he usually pay for its use ? What is the name given to money paid for the use of money ? 4. What is the meaning of the following : '' Money to loan on good securities. Interest 5%." ? 5. A man borrowed $600 for one year. He paid f .05 for the use of each dollar, or 5% of the amount borrowed. How much interest did he pay ? 6. A boy borrowed $40 from his father to buy a bicycle. He agreed to pay his father 5% interest. How much interest should he pay each year ? 7. A contractor borrowed $3000 at 6% and used the money to build a house, which he rented at $20 a month. Find the interest which he must pay each year. Find the amount of rent which he receives each year. How much more does the rent amount to than the interest ? a Find the interest on $1800 for 1 year at 6%. 9. Find the interest on $2000 for 1 year at 8%. 10. Find the interest on $2000 for 2 years at 8%. 11. Find the interest on $6500 for 1 year at ^%. The money borrowed or loaned is called the principal. 238 FRACTIONS AND DECIMALS DIVISION OF DECIMALS 361. Oral Exercises. 1. 4 pt. -f- 2 pt. = — . 8qt.^2qt. = — . 2. 16 gal. -5- 2 gal. = — . 10 mi. -f- 2 mi. = — . 3. 8 tenths -^2 tenths = — . 4 hundredths -^ 2 hundredths = — . 4. .8^.2 = —. ,04-^.02 = —. 5. .12^.06 = —. .8-^.4 = —. 6. .008-^.002 = —. .044-^.004 = —. 7. If the divisor contains tenths, tenths of the dividend may give a whole number in the quotient. 1.6 ft. -^ .2 ft. = 8. 8. If the divisor contains hundredths, hundredths of the dividend may give a whole number in the quo- tient. .16 ft. -^.02 ft. = 8. 362. Written Exercises. 1. Divide 12.2 by .2. 61 . Model: .2)12.2 Note first the lowest order in the divisor. In this case tenths is the lowest order in the divisor. Place the decimal point above and after the figure of the dividend occupying tenths' place. Divide without reference to the decimal point in the quotient. Place the decimal point in the quotient above and after the figure in the dividend occupying the same order or place as the lowest order in the divisor. DIVISION OF DECIMALS 289 2. Divide .66 by .02 ; .328 by .04. Model : .02).66 Model : .04).328 3. Divide 6.6 by .03. 220 . Model: .03)6.60 As the divisor contains hun- dredths, change the dividend to hundredths by an- nexing a cipher. Place the decimal point above and after hundredths' place. Why ? 4. Divide 12 by .002. Model: .002)12.000' 5. Divide .126 by 2. .063 Model: 2.). 126 As units' order is the lowest order in the divisor, place the decimal point above and after the units' place of the dividend. 2 is contained in 1 no times. Write in the quotient. Complete the division. 363. Arrange as in the models and fix the decimal point in the quotient : a 6 c d 1. 1.26^4 .36^.02 3.6^2 360 -> .002 2. .52-^.2 72.8^6 7.2^2 360^7.5 3. 1.55 H- 5 20.65^5 15.5^.05 4.2^32.62 4. 1.6^2 16 H- .002 .16^20 .678-^629 5. 2.4 -^ 6 .240^.006 2.40^12 .54^.0008 240 FRACTIONS AND DECIMALS 364. Written Exercises. Arrange as in the models and fix the decimal point before dividing : 1. Divide 12 by .02; 82 by .04; 5.2 by .02. 2. Divide 3.6 by .2; 7.2 by .2; .72 by .02. 3. Divide 13.6 by .02 ; 6.66 by .002. 4. Divide .155 by 5 ; .2065 by 5. 5. Divide each by .002 : 1.6, 16, .16, .0016. 6. Divide each by 5 : 1.5, .015, .15, .0015. 7. Divide each by .05 : .35, 3.5, .035, .003.5. ^8. Divide each by .025 : 62.5, 625, 6.25, .0625. ' 9. Divide each in No. 8 by 2.5 ; by 25dr ^ 10. Divide each by 9.3 : 23.25, 4.65, 465, .0465. IT. Divide each by $'.96: $240, $2.40, $.48. 12. Divide $170 by $.85 ; 138 by 95 ; 625 by .25. 13. At $.96 each, how many books can be bought for $ 2.40 ? For $ 24 ? For $ 4.80 ? For $ 48 ? 365. Oral Exercises. 1. Find 6% of $350. Find 8^% of $250. 2. Find 71% of $360. Find 9^% of $360. 3. Find 3% of $60; of $27; of $35; of $3.50. 4. Find 21% of $12; of $20; of $30; of $24. 5. Find 8% of $60; of $90; of $75; of $62. 6. 3% of my money is $ 9. Find 1 % of my money. 7. 1% of my money is $3. Find 100% of my money. DIVISION OF DECLMALS 241 366. Oral Exercises. 1. 7 is 1 of 14. 7 is— % of 14. 2. 6 is — third of 18. 6 is — % of 18. 3. 5 is — third of 15. 5 is — % of 15. 4. 7 is — third of 21. 7 is — % of 21. 5. 1 pt. is — half of 1 quart. 1 pt. is — % of 1 quart. 6. 1 qt. is — fourth of 1 gaL 1 qt. is — % of 1 gal. 7. 1 ft. is — third of 1 yd. 1 ft. is — % of 1 yd. 8. 3 in. are — fourth of 12 in. 3 in. are — % of 12 in. 9. 6 in. are of 12 in. 6 in. are — % of 12 in. 10. 2 pt. are of 1 gaL 2 pt. are — % of 1 gal. 11. 5 men are of 15 men. 5 men are — % of 15 men. 12. 6 men are ^ of — men. 6 men are 33 J % of — men. 13. 12 men are f of — men. 12 men are 40 % of — men. • 14. $ 5 is 1 of $— . $5 is 20% of $— . 15. 16 men are ^ oi — men. 16 men are 50 % of — men. 16. A boy sold an article for | of its cost. He gained what part of the cost ? He gained 5 cents. Find the cost. 17. 5% of my money is $25. Find 1% of my money. . Find 100% of my money. Iht Rk a kith— 1(5 242 FRACTIONS AND DECIMALS 367. Written Exercises. 1. Divide 4.628 by 89. ■ Model : .052 Place the decimal point in the 89)4.628 quotient above and after the order 4 45 in the dividend that corresponds 178 to the lowest order in the divisor, 178 — in this case, units. 89 is con- tained in 4 no times. This shows that there is no whole number in the quotient. Do not write the in the quotient. 89 is contained in 46 no times. Write the in tenths' place in the quotient. 462 will contain 89. Complete the division. 2. Divide each by 87: 20.01, 391.5, 7.743, 2.2815. 3< Divide each by 7.9 : 1.6195, 24.174, .32232, 481.11. j4, Divide each by 5.23 : 42.886, .8368, 26^15. 5. A boy paid $ .12 for 6 oranges. Find the cost of 1 orange. 6. A boy paid $ .06 for 12 apples. Find the cost of 1 apple. 7. A farmer sold 220 boxes of apples for $121. Find the selling price per box. 8. Find what per cent 8 is of 16 ; 12 is of 48; 9 is of 72; 64 is of 128. 9. Find what per cent 45 is of 50 ; 40 is of 50 ; 30 is of 20. WRITTEN EXERCISES 248 368. To multiply a number by 25, oS^, 66|, 50, etc., multiply the number by 100 and take such a part of the product as the multiplier is o/" 100. 1. Multiply 624 by 25. V Model : 15600 25 is i of 100. Multiply 624 4)62400 by 100, and take J of the product. 2. Multiply: 78 by 66f ; 69 by 331; 240 by 371; 240 by 621 ; 480 by 75. 3. Multiply: 360 by 25; 1876 by 75; 1728 by 121; 5280 by 331; 144 by 50. 369. To divide a number by 25, 331, 371.^ 66 1^ 75^ etc., divide the numher by 100 and midtiply the residt by the inverted form of the fraction that indicates the part the multiplier is 0/ 100. Model : 14.40 66f is | of 100. Divide 1440 1. Divide 1440 by 66f . 61 3 by 100, and multiply the quotient 2 )43.20 byf. 21.60 2. Divide: 982 by 331; by 75; by 66|; by 621; by 371 3. Divide: 1728 by 75; 5280 by 37i; 5760 by 331 ; 640 by 121 • 6335 by 14f . 244 FRACTIONS AND DECIMALS 370. To divide a number by 200, 300, 2000, etc., point off as tnany places from the rig Jit as there are ciphers in the divisor, and divide by the left-hand figure of the divisor. 1. Divide 627 by 200. Model : 3.135 Point off two places. Divide 2)6.27 6.27 by 2. 2. Divide 768 by 3000. Model : .256 Point off three places. Divide 3).768 .768 by 3. 3. Divide : 5758 by 2000 ; 8520 by 2000 ; 68,960 by 5000. 371. To find 25^, 33^^, 66f^, etc., of a number, take such a part of the number as the required per cent is of 100/^. 1. Find 331^ of $7521. Model : $2507 331^ is J of 100?^ 3)$ 7521 Take 1 of $7521. 2. Find 371^ of $88 ; 331?^ of 66 ; 62^^ of $64 ; 75^0 of $160; 871^ of $64. 3. A man bought a farm for $1200. He sold it at a profit of 33^ ^^ What was his gain? What was his selling price ? 4. $12 is f of what nmnber? $12 is ^^fo more than what number ? CHAPTER VI DENOMINATE NUMBERS 372. Denominate units of measure have been estab- lished in accordance with law, or custom, to measure values, weight, time, length, surface, capacity, etc. UNITED STATES MONEY . 10 mills (m.) = 1 cent (^) 10 cents = 1 dime (d.) 10 dimes = 1 dollar ($) 10 dollars = 1 eagle (E.) How many dollars are there in a double eagle ? Beginning with the one of least value, name the coins that are in circulation. Beginning with the one of least value, name the bills that are in circulation. 373. PAP^R MEASURE 24 sheets = 1 quire 20 quires = 1 ream 2 reams = 1 bundle 5 bundles = 1 bale How many sheets of paper are there in J quire? In 2 quires ? In ^ quire ? 245 246 DENOMINATE NUMBERS 374. COUNTING 12 units = 1 dozen (doz.) 12 dozen = 1 gross 12 gross = 1 great gross 20 units = 1 score Name something that is sold by the gross. 375. TIME MEASURE 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da.) 7 days = 1 week (wk.) 52 weeks = 1 year (yr.) 365 days = 1 year 366 days = 1 leap year 100 years = 1 century. A centennial year is one whose number is divisible by 100. Centennial years whose numbers are divisi- ble by 400, and other years whose numbers are divisible by 4, are leap years. 376. LIQUID MEASURE 4 gills = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 31-| gallons = 1 barrel (bbl.) 2 barrels = 1 hogshead (hhd.) WRITTEN PROBLEMS 247 377. Written Problems. 1. Find the number of years, months, and days from January 3, 1873, to November 1, 1904. Model : 1904 yr. 11 mo. 1 da. November 1, 1873 yr. 1 mo. 3 da. 1 904, is the first 31 yr. 9 mo. 28 da. day of the elev- enth month in the year 1904. January 3, 1873, is the third day of the first month in the year 1873. Write these dates as above, and subtract. 3 da. cannot be taken from 1 da., so add 30 da. (one month) to 1 da. Subtract 3 da. from 31 da. As one month was added to the minuend, add one month to the subtrahend. This changes 1 mo. to 2 mo. Subtract 2 mo. from 11 mo. Then subtract 1873 yr. from 1904 yr. The answer is 31 yr., 9 mo., and 28 da. 2. Abraham Lincoln w^as born February 12, 1809. He died April 15, 1865. How old was he when he died? 3. George Washington died December 14, 1799. How many years is it since his death ? 4. Daniel Webster died October 24, 1852, at the age of 70 yr. 9 mo. 6 da. What was the date of his birth ? 5. William Penn was bom October 14, 1644. He died at the age of 73 yr. 9 mo. 16 da. What was the date of his death ? 6. A man borrowed money January 16, 1902. He paid it March 7, 1903. How long did he keep the money? 248 DENOMINATE NUMBERS 378. AVOIRDUPOIS WEIGHT 16 ounces (oz.) = 1 pound (lb.) 100 pounds = 1 hundredweight (cwt.) 2000 pounds = 1 ton (T.) The long ton contains 2240 pounds. It is used at the custom-houses in invoices of some imports, and sometimes in weighing coal. 379. Oral Exercises. 1. Hold up enough books to weigh about one pound. 2. An ounce is what part of a pound ? 3. How many hundredweight are there in 600 lb. ? In 624 lb. ? In 52 lb. ? In 167 lb. ? 4. How many pounds are there in ^ cwt. ? In 2^ cwt. ? In 3f cwt. ? In IT.? In 1 T. and 2 cwt. ? 380. Written Exercises. 1. A bushel of wheat weighs 60 lb. How many bushels of wheat will weigh IT.? 3J T. ? 5.4 T. ? 2. When wheat is worth $.80 a bushel, how much is 1 T. worth ? 3. When wheat is selling at $1.75 per hundred- weight, what is the price per bushel ? 4. A farmer sold his wheat at $18 a ton. He liad 63,896 lb. of wheat. How much did he receive for his crop ? 5. What per cent of a pound is 8 ounces ? MEASURE OF LENGTH 249 MEASURE OF LENGTH lyd. i_ 1 u I I I I 1 I I I I I I 1 1 ft. 12 in. 381. 1. One foot is what part of 1 yd. ? 2. One inch is what part of 1 ft. ? It is what per cent of 1 ft. ? 3. What part of 1 ft. are 4 in. ? 6 in. ? 9 in. ? 8 in. ? 2 in. ? 3 in. ? 4. What per cent of 1 ft. are 4 in. ? 6 in. ? 9 in. ? 8 in.? 2 in.? 3 in.? 12 in.? 5. A fathom (6 ft.) is used to measure the depth of the sea. 6. A chain (4 rd.) is used hy surveyors in measur- ing land. 7. A hand (4 in.) is used in measuring the height of a horse. 8. A horse is 15 hands high. How many feet high is the horse ? SQUARE MEASURE 382. See p. 175. 1. Find the number of square feet and the number of square yards in a surface 36 ft. long and 24 ft. wide. 2. How many square feet are there in the floor of your schoolroom ? 3. Find the number of square yards of plastering there are in your schoolroom. 250 DENOMINATE NUMBERS CUBIC MEASURE 383. 1. A cubic inch is a solid whose six equal sidea are each 1 sq. in. 2. A cubic foot is a solid whose six equal sides are each 1 sq. ft. 3. Examine the cubic inch. Has it length ? Has it width ? Has it thickness ? 4. How many cubic inches will cover a square foot of surface ? Try it. 5. If you make them 2 deep, how many cubic inches can you place on a square foot of surface ? 3 deep? 12 deep? The pile 12 deep will be a cubic foot. 7. There are — cubic inches in a cubic foot. 8. Think of a box 8 in. long, 4 in. wide, and 3 in. high. How many cubic inches could be placed on the bottom of the box ? How many deep could you place the blocks before the box would be full ? Model : 8 cu. in. = the number of cubic inches that can be placed in a space 8 in. long, 1 in. wide, and 1 in. high. 32 cu. in. = the number of cubic inches that can be placed in a space 8 in. long, 4 in. wide, and 1 in. high. 96 cu. in. = the number of cubic inches in a space 8 iiL long, 4 in. wide, and 3 in. high. CUBIC MEASURE 251 384. 1. Memorize the following : 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd^) 128 cubic feet = 1 cord of wood 2. Draw a cubic inch. Draw a cubic foot. 3. How can you find how many cubic inches there are in 2 cu. ft.? 4. How many cubic inches are there in 3 cu. ft. ? In 3 cu. ft. and 120 cu. in.? 5. A man dug a cellar 18 ft. long, 12 ft. wide, and 6 ft. deep. How many cubic feet of earth did he remove ? How many cubic yards did he remove ? He was paid $.32 a cubic yard. How much did he receive for the work? 6. A trench was dug 3 ft. 6 in. (3J ft.) wide and 12 ft. deep, in which to lay a sewer. The sewer w^as 1 mile (5,280 ft.) long. How many cubic yards of earth were removed ? 7. Find the number of cubic feet in a space 12 ft. long, 8 ft. wide, and 6 ft. deep. 8. Find the number of cubic feet in your school- room. 9. A box contains 60 cu. ft. It is 5 ft. long and 4 ft. high. How wide is it ? 10. A room contains 1620 cu. ft. It is 15 ft. long and 12 ft. wide. How high is it? 252 DENOMINATE NUMBERS LUMBER MEASURE 385. Lumber is measured by the board foot. A board foot is a piece of lumber one foot long, one foot wide, and one inch thick. 7b find the number of hoard feat in a piece of lumber, multiply the length of the piece in feet by the thickness of the- piece in inches, and this by the tvidth of the piece in inches, and divide the product by 12. To shorten the work use cancellation. 1. Find the number of feet of lumber in a piece of lumber 16 it. long, 9 in. wide, and 3 in. thick. 4 Model : J^ x9x? 36, the number of board feet. 2. Find the number of feet of lumber in 12 pieces, each 16 ft. long, 8 in. wide, and 1 in. thick. 3. Find the number of board feet in a timber 20 ft. long, 8 in. wide, and 8 in. thick. 4. Find the number of board feet of flooring in the floor of your schoolroom. 5. At $ 12 per thousand feet, what will be the cost of 20 pieces, each 10 ft. long, 4 in. wide, and 2 in. thick ? 6. Find the cost of lumber at a neighboring lumber yard. CASH ACCOUNT CASH ACCOUNT 386. A cash account is a written statement of cash received and cash paid out. Dr. Keceived cash. Paid out. Cr, 1905 1905 Jan. 1 On hand 8 75 Jan. 2 By cash 4 00 a 8 To salary 12 00 a 13 " board 7 00 (( 16 To remit 30 00 i a 16 " bank deposit 30 00 11 24 To salary 12 62 00 1 75 a 25 Balance 21 62 1^ Jan. 25 On hand 21 75 387. 1. The left-hand side, or debit side, of a cash account shows the cash received and from what sources it was received. 2. What does the right-hand, or credit side, of a cash account show ? 3. Why should one keep a cash account ? 4. What is meant by the entry " On hand $8.75 " ? 5. The above was Mr. A's cash account from January 1 to January 25, 1905. How much cash did Mr. A receive during this time ? 6. With the $8.75, how much cash must be ac- counted for ? 7. What did Mr. A do with his money ? a How much money had Mr. A on hand Jan. 25 ? 9. What is meant by *' Balance " ? 254 DENOMINATE NUMBERS 10. This was Harry's cash business for the month of February, 1904. Rule your paper, make up, and close Harry's account. Feb. 1 Harry haoy on hand $.45; Feb. 2 he paid out for papers $.35, ^d received for papers $.70 ; Feb. 3 he received for weeding a garden ("for labor") $.60; Feb. 4 he paid for papers $.60, and received for papers $1.2(rpFeb. 8 he^~-paid $.10 carfare, and received for delivering a package $.35; Feb. 9 he bought a book for his sister, pay- ing $.20; Feb. 12 he earned $1, and spent for car- fare $.20. 11. Make similar accounts. ANGLES 388. An angle is the opening between two lines that meet. Angle Right Angle Acute Angle Obtuse Angle 1. Join two lines at a point not at the ends of the lines. How many right angles is it possible to make with two lines thus joined? How many obtuse angles ? How many acute angles ? 2. A right angle is an angle formed by the meet- ing of one straight line perpendicular to another. 3. An acute angle is an angle that is less than a right angle. ANGLES 255 4. An obtuse angle is an angle that is greater than a right angle. 5. How many angles has a square ? An oblong ? What kind of angles are they ? 6. Draw a circle on the blackboard. Divide it into fourths. How many right angles are there in the circle? 7. In the figures on page 64, which angles are right angles ? 8. The line that bounds a circle is called its circumference. 9. Angles are nieasured in degrees. The angles of a circle are measured on its circumference. There are 360 degrees in a circle. 10. How many degrees are there in a right angle ? In one half of a right angle ? 11. Divide a right angle into three equal angles. How many degrees are there in each of these angles ? 12. An angle of -180 degrees is equal to two right angles. Explain why there are 180 degrees from the north pole to the south pole. 13. Explain the use of meridians and parallels. 389. CIRCULAR MEASURE 60 seconds ('') = 1 minute (') 60 minutes = 1 degree (°) 360 degrees = 1 circle 256 ROMAN NOTATION 390. ROMAN NOTATION 1 I 10 X lOO C 1000 M 2 II 20 XX 200 CC 2000 MM 3 III 30 XXX 300 CCC 3000 MMM 4 IV 40 XL 400 CD 4000 IV 5 V 50 L 500 D 5000 V 6 VI 60 LX 000 DC 6000 VI 7 VII 70 LXX 700 DCC 7000 VII 8 VIII 80 LXXX 800 DCCC 8000 VIII 9 IX 90 xc 900 CM 9000 IX 391. 1. The letters used in Roman notation are : I, V, X, L, C, D, M. 2. When a letter of less value is written before a letter of greater value, its value is taken from that of the letter of greater value ; as, IV, IX, XL. When it is. placed after a letter of greater value, it is added; as, VI, XI, LV, etc. 3. A dash placed over a letter or a combination of letters increases the value a thousand-fold. 4. Write in Roman notation : 27, 34, 68, 89, 235, 309, 540, 894, 1000. 5. Write in Roman notation: 1890, 1776, 1904, 2000. 6. Write in figures : CDXXI MXCIV MDCCCXCIX 7. Where have you seen the Roman numerals used ? YB 35856 ivil8750J2 ■=':> THE UNIVERSITY OF CALIFORNIA LIBRARY